http://www.polyfolds.org/api.php?action=feedcontributions&user=Natebottman&feedformat=atomPolyfolds.org - User contributions [en]2024-03-29T11:13:29ZUser contributionsMediaWiki 1.24.4http://www.polyfolds.org/index.php?title=Main_Page&diff=631Main Page2017-06-17T18:46:38Z<p>Natebottman: </p>
<hr />
<div><center><br />
[[File:poly_group.jpeg | 1200px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
** [[https://drive.google.com/drive/folders/0B2B4teKY2S2PMDR6blZEcDN6cE0?usp=sharing|notes Jeff's notes]]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Main_Page&diff=630Main Page2017-06-17T18:46:26Z<p>Natebottman: </p>
<hr />
<div><center><br />
[[File:poly_group.jpeg | 1000px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
** [[https://drive.google.com/drive/folders/0B2B4teKY2S2PMDR6blZEcDN6cE0?usp=sharing|notes Jeff's notes]]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=File:Poly_group.jpeg&diff=629File:Poly group.jpeg2017-06-17T18:46:09Z<p>Natebottman: </p>
<hr />
<div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Main_Page&diff=628Main Page2017-06-17T18:45:37Z<p>Natebottman: </p>
<hr />
<div><center><br />
[[File:poly_group.jpg | 1000px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
** [[https://drive.google.com/drive/folders/0B2B4teKY2S2PMDR6blZEcDN6cE0?usp=sharing|notes Jeff's notes]]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Links_to_Videos,_Papers,_and_Ongoing_Work_on_Polyfold_Theory&diff=621Links to Videos, Papers, and Ongoing Work on Polyfold Theory2017-06-14T18:22:48Z<p>Natebottman: /* Videos of talks */</p>
<hr />
<div>== Videos of talks ==<br />
<br />
* 2015 Summer School on Moduli Problems in Symplectic Geometry playlist [https://www.youtube.com/playlist?list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3], in particular series by J.Fish, K.Wehrheim [https://www.youtube.com/watch?v=65OmPRO5-jA&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3&index=36], [https://www.youtube.com/watch?v=dQv-QlbyExs&index=33&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3], [https://www.youtube.com/watch?v=M0gCG3gOYds&index=30&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3], [https://www.youtube.com/watch?v=SOdBRwqqHSQ&index=28&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3], [https://www.youtube.com/watch?v=-Wpmxymnvk0&index=21&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3]; discussions with N.Bottman [https://www.youtube.com/watch?v=JuoahjHz7gs&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3&index=26], [https://www.youtube.com/watch?v=J2wKKEAVrZ4&index=15&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3]; H.Hofer on construction of SFT polyfolds [https://www.youtube.com/watch?v=IVVLPF3r3_w&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3&index=18], [https://www.youtube.com/watch?v=sAjxDmhgHqc&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3&index=12], [https://www.youtube.com/watch?v=47b0k5lLB7c&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3&index=8], [https://www.youtube.com/watch?v=2KYFTcGbLps&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3&index=5], [https://www.youtube.com/watch?v=fjW_WL3KLUg&list=PLx5f8IelFRgGaUFEBurqstanRzuFFUlT3&index=1]<br />
<br />
* Lecture course [[https://math.berkeley.edu/~katrin/teach/regularization/vlectures.shtml Regularization Of Moduli Spaces Of Pseudoholomorphic Curves]] (K.Wehrheim, 2014)<br />
<br />
* Introduction to Polyfolds (K.Wehrheim, 2012 at IAS) [https://video.ias.edu/intropolyfolds/wehrheim]<br />
<br />
* Lecture series on Polyfolds (H.Hofer, 2012 at IAS) [https://video.ias.edu/polyfoldsminicourse/hofer1], [https://video.ias.edu/polyfoldsminicourse/hofer2], [https://video.ias.edu/polyfoldsminicourse/hofer3], [https://video.ias.edu/polyfoldsminicourse/hofer4], [https://video.ias.edu/polyfoldsminicourse/hofer5], [https://video.ias.edu/polyfoldsminicourse/hofer6], [https://video.ias.edu/polyfoldsminicourse/hofer7], [https://video.ias.edu/polyfoldsminicourse/hofer8] <br />
<br />
* An M-polyfold relevant to Morse theory (P.Albers, 2012 at IAS) [https://video.ias.edu/intropolyfolds/albers]<br />
<br />
* Transversality questions and polyfold structures for holomorphic disks (K.Wehrheim, 2009 at MSRI) [https://www.youtube.com/watch?v=4kgiFIkS-f0&list=PLXjpiiAr8QGJODAieckuuzks6fuRAKKLB&index=20]<br />
<br />
== Surveys and Textbooks == <br />
<br />
* A Polyfold Cheat Sheet (K.Wehrheim, 2016) [https://math.berkeley.edu/~katrin/polylab/cheat.pdf]<br />
<br />
* Polyfold and Fredholm Theory I: Basic Theory in M-Polyfolds (H.Hofer, K.Wysocki, E.Zehnder, 2014) [https://arxiv.org/abs/1407.3185]<br />
<br />
* Polyfolds And A General Fredholm Theory (H.Hofer, 2008&2014) [https://arxiv.org/abs/1412.4255]<br />
<br />
* Polyfolds: A First and Second Look (O.Fabert, J.Fish, R.Golovko, K.Wehrheim, 2012) [https://arxiv.org/abs/1210.6670]<br />
<br />
* A General Fredholm Theory and Applications (H.Hofer, 2005) [https://arxiv.org/abs/math/0509366]<br />
<br />
== Abstract Polyfold Theory ==<br />
<br />
* A General Fredholm Theory I: A Splicing-Based Differential Geometry (H.Hofer, K.Wysocki, E.Zehnder, 2006) [https://arxiv.org/abs/math/0612604]<br />
<br />
* A General Fredholm Theory II: https://arxiv.org/abs/0705.1310 (H.Hofer, K.Wysocki, E.Zehnder, 2007) [https://arxiv.org/abs/0705.1310] <br />
<br />
* A General Fredholm Theory III: Fredholm Functors and Polyfolds (H.Hofer, K.Wysocki, E.Zehnder, 2008) [https://arxiv.org/abs/0810.0736]<br />
<br />
* Integration Theory for Zero Sets of Polyfold Fredholm Sections (H.Hofer, K.Wysocki, E.Zehnder, 2007) [https://arxiv.org/abs/0711.0781]<br />
<br />
* Sc-Smoothness, Retractions and New Models for Smooth Spaces (H.Hofer, K.Wysocki, E.Zehnder, 2010) [https://arxiv.org/abs/1002.3381]<br />
<br />
* ''Coherent M-polyfold Theory'' (B.Filippenko, K.Wehrheim) - provides a construction scheme for perturbations of Fredholm sections (with trivial isotropy) whose boundary stratifications are Cartesian products of other Fredholm sections, while preserving transversality and compactness; preliminary draft available<br />
<br />
* ''Sliceable Polyfolds and Constrained Moduli Problem'' (B.Filippenko, K.Wehrheim) - establishes an implicit function theorem for non-Fredholm submersions cutting out 'slices' of finite codimension, proves that restrictions of polyfold Fredholm sections to 'slices' remain Fredholm, and in particular proves that suitably transverse fiber products of Polyfold Fredholm sections are again Polyfold Fredholm; preliminary draft available<br />
<br />
* ''Quotients in Polyfold Theory'' (Z.Zhou, K.Wehrheim) - studies Polyfold Fredholm sections that are equivariant under the action of a compact Lie group, constructs equivariant transverse perturbations for actions with finite isotropy and <math>S^1</math> actions with vanishing obstructions, and applies this to prove Arnold conjecture; preliminary draft available<br />
<br />
* ''Equivariant Fundamental Class and Localization Theorem'' (Z.Zhou, K.Wehrheim) - constructs an equivariant fundamental class for any equivariant polyfold Fredholm section (without boundary), and proves the localization theorem for 'regular fixed point locus'; preliminary draft available<br />
<br />
== Applications of Polyfold Theory ==<br />
<br />
* Applications of Polyfold Theory I: The Polyfolds of Gromov-Witten Theory (H.Hofer, K.Wysocki, E.Zehnder, 2011) [https://arxiv.org/abs/1107.2097]<br />
<br />
* Fredholm notions in scale calculus and Hamiltonian Floer theory (K.Wehrheim, 2012&2016) [https://arxiv.org/abs/1209.4040]<br />
<br />
* A-infty structures from Morse trees with pseudoholomorphic disks (Jiayong Li, K.Wehrheim, 2014 preliminary draft) [https://math.berkeley.edu/~katrin/papers/disktrees.pdf]<br />
<br />
* ''Applications of Polyfold Theory II: The Polyfolds of SFT'' (J.Fish, H.Hofer) - constructs Polyfold Fredholm sections whose zero sets are the SFT moduli spaces<br />
<br />
* ''SFT via Polyfold Theory'' (J.Fish, H.Hofer) - constructs the SFT algebra package via coherent perturbations and a study of homotopies of data<br />
<br />
* ''A proof of the Arnold Conjuecture by Polyfold Techniques'' (P.Albers, B.Filippenko, J.Fish, K.Wehrheim) - constructs the PSS isomorphism in general compact symplectic manifolds - preliminary draft and [[https://math.berkeley.edu/~katrin/slides/arnold.pdf slides]] available<br />
<br />
* ''Gromov-Witten axioms for polyfold construction'' (W.Schmaltz, K.Wehrheim) - proves the Kontsevich-Manin axioms for the polyfold construction of Gromov-Witten invariants; writing in progress<br />
<br />
* ''Polyflow Homotopy Theory'' (Z.Zhou) - develops homotopical invariants from systems of Polyfold Fredholm sections whose zero sets- if cut out transversely - form a Cohen-Jones-Segal 'flow category', applies in Morse-Bott settings, and provides new approaches to proving invariance of algebraic structures under homotopies of data; work in progress</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Main_Page&diff=620Main Page2017-06-14T05:43:39Z<p>Natebottman: /* Announcements */</p>
<hr />
<div><center><br />
[[File:polyfold_machine.png | 600px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
** [[https://drive.google.com/drive/folders/0B2B4teKY2S2PMDR6blZEcDN6cE0?usp=sharing|notes Jeff's notes]]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Main_Page&diff=619Main Page2017-06-14T05:43:18Z<p>Natebottman: /* Contents */</p>
<hr />
<div><center><br />
[[File:polyfold_machine.png | 600px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Main_Page&diff=618Main Page2017-06-13T23:04:55Z<p>Natebottman: /* Contents */</p>
<hr />
<div><center><br />
[[File:polyfold_machine.png | 600px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]], [[https://drive.google.com/drive/folders/0B2B4teKY2S2PMDR6blZEcDN6cE0?usp=sharing|notes Jeff's notes]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Main_Page&diff=617Main Page2017-06-13T23:04:38Z<p>Natebottman: /* Contents */</p>
<hr />
<div><center><br />
[[File:polyfold_machine.png | 600px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]], [[https://drive.google.com/drive/folders/0B2B4teKY2S2PMDR6blZEcDN6cE0?usp=sharing|notes by Jeff]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Main_Page&diff=616Main Page2017-06-13T23:04:25Z<p>Natebottman: /* Contents */</p>
<hr />
<div><center><br />
[[File:polyfold_machine.png | 600px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]], [[https://drive.google.com/drive/folders/0B2B4teKY2S2PMDR6blZEcDN6cE0?usp=sharing|Jeff's notes]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Main_Page&diff=615Main Page2017-06-13T23:04:12Z<p>Natebottman: /* Contents */</p>
<hr />
<div><center><br />
[[File:polyfold_machine.png | 600px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]], [[https://drive.google.com/drive/folders/0B2B4teKY2S2PMDR6blZEcDN6cE0?usp=sharing | Jeff's notes]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Main_Page&diff=614Main Page2017-06-13T23:03:51Z<p>Natebottman: /* Contents */</p>
<hr />
<div><center><br />
[[File:polyfold_machine.png | 600px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]], [[https://drive.google.com/drive/folders/0B2B4teKY2S2PMDR6blZEcDN6cE0?usp=sharing|Jeff's notes]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Main_Page&diff=613Main Page2017-06-13T23:03:34Z<p>Natebottman: /* Contents */</p>
<hr />
<div><center><br />
[[File:polyfold_machine.png | 600px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]], [[<br />
https://drive.google.com/drive/folders/0B2B4teKY2S2PMDR6blZEcDN6cE0?usp=sharing|Jeff's notes]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Main_Page&diff=612Main Page2017-06-13T23:03:11Z<p>Natebottman: /* Contents */</p>
<hr />
<div><center><br />
[[File:polyfold_machine.png | 600px]]<br />
</center><br />
<br />
== Contents ==<br />
<br />
* [[Links to Videos, Papers, and Ongoing Work on Polyfold Theory]]<br />
* [[Polyfold Constructions for Fukaya Categories]] - a crowd sourcing project - [[table of contents]], [[<br />
https://drive.google.com/drive/folders/0B2B4teKY2S2PMDR6blZEcDN6cE0?usp=sharing]]<br />
* list of contributors: [[polyfold lab]]<br />
<br />
== Announcements ==<br />
<br />
* Polyfold Theory towards the Fukaya Category, June 12-16, 2017 at UC Berkeley [https://math.berkeley.edu/~katrin/summer/] <br />
** [[RTG workshop program]]<br />
** [https://piazza.com/configure-classes/other/polyfoldsummer e-discussion forum (piazza)]<br />
<br />
* SFT 9 Conference August 27-31 in Augsburg, Germany ... stay tuned!<br />
<br />
== Getting started ==<br />
* Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [//www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Problems_on_Deligne-Mumford_spaces&diff=598Problems on Deligne-Mumford spaces2017-06-13T00:33:52Z<p>Natebottman: /* More low-dimensional examples of Deligne-Mumford spaces */</p>
<hr />
<div>These problems deal with Deligne-Mumford spaces, by which we mean the moduli spaces of domains relevant for our construction of the Fukaya category.<br />
The first problem forms the '''warm-up portion''': you should make sure you understand how to do this one before Tuesday morning.<br />
The remaining two problems form the '''further fun section''': useful for deeper understanding, but not essential for following the thread of the lectures.<br />
<br />
== Down and dirty with low-dimensional associahedra ==<br />
<br />
Using the notation of [[Deligne-Mumford space]], for any <math>d\geq 2</math>, the '''associahedron''' <math>\overline\mathcal{M}_{d+1} := DM(d,0,(\{0\},\ldots,\{d\}))</math> is a <math>(d-2)</math>-dimensional manifold with boundary and corners which parametrizes nodal trees of disks with <math>d+1</math> marked points, one of them distinguished (we think of the <math>d</math> undistinguished resp. 1 distinguished marked points as "input" resp. "output" marked points).<br />
The associahedra are one of the Deligne-Mumford spaces we will use during the summer school, corresponding to the situation where all the Lagrangians labeling the boundary segments are distinct.<br />
<br />
'''(a)''' As shown in [Auroux, Ex. 2.6 [[https://arxiv.org/pdf/1301.7056.pdf]]], <math>\overline\mathcal{M}_4</math> is homeomorphic to a closed interval, with one endpoint corresponding to a collision of the first two inputs <math>z_1, z_2</math> and the other corresponding to a collision of <math>z_2, z_3</math>.<br />
Moving up a dimension, <math>\overline\mathcal{M}_5</math> is a pentagon; it can be identified with the central pentagon in the depiction of <math>DM(4,0;(\{0,1,2,3,4\}))</math> in [[Deligne-Mumford space]].<br />
Which polyhedron is <math>\overline\mathcal{M}_6</math> equal to?<br />
(A good way to get started on this problem is to list the codimension-1 strata.)<br />
<br />
'''(b)''' Using the manifold-with-corners structure of the associahedra constructed in [[Deligne-Mumford space]], observe that the <math>1</math>-dimensional associahedron <math>\overline\mathcal{M}_4</math> can be covered by three charts:<br />
* boundary charts centered respectively at the two points in <math>\overline\mathcal{M}_4 \setminus \mathcal{M}_4</math> (the domains of these charts are of the form <math>[0,a)</math>, and a choice of a point in this interval tells us how much to smooth the node);<br />
* an interior chart (which we produce by fixing the positions of three of the marked points, and varying the position of the fourth).<br />
Explicitly work out these charts, and the transition maps amongst them.<br />
<br />
== The poset indexing the strata of the associahedra ==<br />
<br />
The associahedron <math>\overline\mathcal{M}_{d+1}</math> can be given the structure of a stratified space, where the underlying poset is called <math>K_d</math> and consists of '''stable rooted ribbon trees with <math>d</math> leaves'''.<br />
Similarly to the setup in [[Moduli spaces of pseudoholomorphic polygons]], a stable rooted ribbon tree is a tree <math>T</math> satisfying these properties:<br />
* <math>T</math> has <math>d</math> leaves and 1 root (in our terminology, the root has valence 1 but is not counted as a leaf);<br />
* <math>T</math> is stable, i.e. every main vertex (=neither a leaf nor the root) has valence at least 3;<br />
* <math>T</math> is a ribbon tree, i.e. the edges incident to any vertex are equipped with a cyclic ordering.<br />
To define the partial order, we declare <math>T' \leq T</math> if we can contract some of the interior edges in <math>T'</math> to get <math>T</math>; we declare that <math>T'</math> is in the closure of <math>T</math> if <math>T'\leq T</math>.<br />
Write the closure of the stratum corresponding to <math>T</math> as a product of lower-dimensional <math>K_d</math>'s.<br />
Which tree corresponds to the top stratum of <math>\overline\mathcal{M}_{d+1}</math>?<br />
To the codimension-1 strata of <math>\overline\mathcal{M}_{d+1}</math>?<br />
<br />
...and, to the operadically initiated (or willing to dig around a little at [[http://ncatlab.org]]): show that the collection <math>(K_d)_{d\geq 2}</math> can be given the structure of an '''operad''' (which is to say that for every <math>d, e \geq 2</math> and <math>1 \leq i \leq d</math> there is a composition operation <math>\circ_i\colon K_d \times K_e \to K_{d+e-1}</math> which splices <math>T_e \in K_e</math> onto <math>T_d \in K_d</math> by identifying the outgoing edge of <math>T_e</math> with the <math>i</math>-th incoming edge of <math>T_d</math>, and that these operations satisfy some coherence conditions).<br />
Next, show that algebras / categories over the operad <math>(C_*(K_d))_{d\geq2}</math> of cellular chains on <math>K_d</math> are the same thing as <math>A_\infty</math> algebra / categories.<br />
<br />
== More low-dimensional examples of Deligne-Mumford spaces ==<br />
<br />
'''(a)''' Explicitly work out the 3-dimensional Deligne-Mumford space <math>DM(3,1;(\{0\},\{1\},\{2\},\{3\}))</math>.<br />
<br />
'''(b)''' For <math>d \geq 2</math> define <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C}) := DM(d)^{\text{sph,ord}}</math>.<br />
That is, if we define<br />
<center><math><br />
\mathcal{M}_{0,d+1}(\mathbb{C}) := \bigl\{ \underline{z} = (z_0,\ldots,z_d)\in \mathbb{CP}^1 \;\text{pairwise disjoint} \bigr\}/_\sim,<br />
</math></center><br />
where two configurations are identified if one can be taken to the other by a Moebius transformation, then <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C})</math> is the compactification defined by including stable trees of spheres, where every sphere has at least 3 special (marked/nodal) points and any neighboring pair of spheres is attached at a pair of points.<br />
(A detailed construction of <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C})</math> can be found in [big McDuff-Salamon, App. D].)<br />
Make the identifications (don't worry too much about rigor) <math>\overline\mathcal{M}_{0,4}(\mathbb{C}) \cong \mathrm{pt}</math>, <math>\overline\mathcal{M}_{0,5}(\mathbb{C}) \cong \mathbb{CP}^1</math>, and <math>\overline\mathcal{M}_{0,6}(\mathbb{C}) \cong (\mathbb{CP}^1\times\mathbb{CP}^1) \# 3\overline{\mathbb{CP}}^2</math>.</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Problems_on_Deligne-Mumford_spaces&diff=593Problems on Deligne-Mumford spaces2017-06-12T13:38:05Z<p>Natebottman: /* Down and dirty with low-dimensional associahedra */</p>
<hr />
<div>These problems deal with Deligne-Mumford spaces, by which we mean the moduli spaces of domains relevant for our construction of the Fukaya category.<br />
The first problem forms the '''warm-up portion''': you should make sure you understand how to do this one before Tuesday morning.<br />
The remaining two problems form the '''further fun section''': useful for deeper understanding, but not essential for following the thread of the lectures.<br />
<br />
== Down and dirty with low-dimensional associahedra ==<br />
<br />
Using the notation of [[Deligne-Mumford space]], for any <math>d\geq 2</math>, the '''associahedron''' <math>\overline\mathcal{M}_{d+1} := DM(d,0,(\{0\},\ldots,\{d\}))</math> is a <math>(d-2)</math>-dimensional manifold with boundary and corners which parametrizes nodal trees of disks with <math>d+1</math> marked points, one of them distinguished (we think of the <math>d</math> undistinguished resp. 1 distinguished marked points as "input" resp. "output" marked points).<br />
The associahedra are one of the Deligne-Mumford spaces we will use during the summer school, corresponding to the situation where all the Lagrangians labeling the boundary segments are distinct.<br />
<br />
'''(a)''' As shown in [Auroux, Ex. 2.6 [[https://arxiv.org/pdf/1301.7056.pdf]]], <math>\overline\mathcal{M}_4</math> is homeomorphic to a closed interval, with one endpoint corresponding to a collision of the first two inputs <math>z_1, z_2</math> and the other corresponding to a collision of <math>z_2, z_3</math>.<br />
Moving up a dimension, <math>\overline\mathcal{M}_5</math> is a pentagon; it can be identified with the central pentagon in the depiction of <math>DM(4,0;(\{0,1,2,3,4\}))</math> in [[Deligne-Mumford space]].<br />
Which polyhedron is <math>\overline\mathcal{M}_6</math> equal to?<br />
(A good way to get started on this problem is to list the codimension-1 strata.)<br />
<br />
'''(b)''' Using the manifold-with-corners structure of the associahedra constructed in [[Deligne-Mumford space]], observe that the <math>1</math>-dimensional associahedron <math>\overline\mathcal{M}_4</math> can be covered by three charts:<br />
* boundary charts centered respectively at the two points in <math>\overline\mathcal{M}_4 \setminus \mathcal{M}_4</math> (the domains of these charts are of the form <math>[0,a)</math>, and a choice of a point in this interval tells us how much to smooth the node);<br />
* an interior chart (which we produce by fixing the positions of three of the marked points, and varying the position of the fourth).<br />
Explicitly work out these charts, and the transition maps amongst them.<br />
<br />
== The poset indexing the strata of the associahedra ==<br />
<br />
The associahedron <math>\overline\mathcal{M}_{d+1}</math> can be given the structure of a stratified space, where the underlying poset is called <math>K_d</math> and consists of '''stable rooted ribbon trees with <math>d</math> leaves'''.<br />
Similarly to the setup in [[Moduli spaces of pseudoholomorphic polygons]], a stable rooted ribbon tree is a tree <math>T</math> satisfying these properties:<br />
* <math>T</math> has <math>d</math> leaves and 1 root (in our terminology, the root has valence 1 but is not counted as a leaf);<br />
* <math>T</math> is stable, i.e. every main vertex (=neither a leaf nor the root) has valence at least 3;<br />
* <math>T</math> is a ribbon tree, i.e. the edges incident to any vertex are equipped with a cyclic ordering.<br />
To define the partial order, we declare <math>T' \leq T</math> if we can contract some of the interior edges in <math>T'</math> to get <math>T</math>; we declare that <math>T'</math> is in the closure of <math>T</math> if <math>T'\leq T</math>.<br />
Write the closure of the stratum corresponding to <math>T</math> as a product of lower-dimensional <math>K_d</math>'s.<br />
Which tree corresponds to the top stratum of <math>\overline\mathcal{M}_{d+1}</math>?<br />
To the codimension-1 strata of <math>\overline\mathcal{M}_{d+1}</math>?<br />
<br />
...and, to the operadically initiated (or willing to dig around a little at [[http://ncatlab.org]]): show that the collection <math>(K_d)_{d\geq 2}</math> can be given the structure of an '''operad''' (which is to say that for every <math>d, e \geq 2</math> and <math>1 \leq i \leq d</math> there is a composition operation <math>\circ_i\colon K_d \times K_e \to K_{d+e-1}</math> which splices <math>T_e \in K_e</math> onto <math>T_d \in K_d</math> by identifying the outgoing edge of <math>T_e</math> with the <math>i</math>-th incoming edge of <math>T_d</math>, and that these operations satisfy some coherence conditions).<br />
Next, show that algebras / categories over the operad <math>(C_*(K_d))_{d\geq2}</math> of cellular chains on <math>K_d</math> are the same thing as <math>A_\infty</math> algebra / categories.<br />
<br />
== More low-dimensional examples of Deligne-Mumford spaces ==<br />
<br />
'''(a)''' Explicitly work out the 3-dimensional Deligne-Mumford space <math>DM(3,1;(\{0\},\{1\},\{2\},\{3\}))</math>.<br />
<br />
'''(b)''' For <math>d \geq 2</math> define <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C}) := DM(d)^{\text{sph,ord}}</math>.<br />
That is, if we define<br />
<center><math><br />
\mathcal{M}_{0,d+1}(\mathbb{C}) := \bigl\{ \underline{z} = \{z_0,\ldots,z_d\}\in \mathbb{CP}^1 \;\text{pairwise disjoint} \bigr\}/_\sim,<br />
</math></center><br />
where two configurations are identified if one can be taken to the other by a Moebius transformation, then <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C})</math> is the compactification defined by including stable trees of spheres, where every sphere has at least 3 special (marked/nodal) points and any neighboring pair of spheres is attached at a pair of points.<br />
(A detailed construction of <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C})</math> can be found in [big McDuff-Salamon, App. D].)<br />
Make the identifications (don't worry too much about rigor) <math>\overline\mathcal{M}_{0,4}(\mathbb{C}) \cong \mathrm{pt}</math>, <math>\overline\mathcal{M}_{0,5}(\mathbb{C}) \cong \mathbb{CP}^1</math>, and <math>\overline\mathcal{M}_{0,6}(\mathbb{C}) \cong (\mathbb{CP}^1\times\mathbb{CP}^1) \# 3\overline{\mathbb{CP}}^2</math>.</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=589Deligne-Mumford space2017-06-10T22:37:01Z<p>Natebottman: /* Some notation and examples */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have surjections<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}} \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points (there is an <math>S_\ell</math>-action on the domain, and this map takes orbits).<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v})</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above). For any vertex <math>v \in \#_{\underline r}(\hat V)</math>, we denote by <math>\hat V^g_v\subset \hat V</math> the vertices in <math>\hat T</math> which are identified to form <math>v</math>. These vertices form a subtree of <math>\hat T</math>; denote the edges in this subtree by <math>\hat E^g_{\underline r}</math>.<br />
* To define the objects <math>\#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)</math>, we will "glue" certain disks and spheres together, using the choices of strip coordinates and cylinder coordinates made above. The crucial constructions will be, for each main vertex <math>v \in \#_{\underline r}(\hat V)</math>, a glued disk <math>\#_{\underline r}(\underline D)_v</math> formed by gluing together the disks corresponding to the vertices in <math>\hat E^g_v</math>. We now make a few definitions towards the construction of <math>\#_{\underline r}(\underline D)_v</math>:<br />
** For <math>\hat e \in \hat E^g_v</math>, define the '''gluing length''' <math>R_{\hat e} := R(r_{\hat e})</math>.<br />
** Denote by <math>L_{R_{\hat e}}</math> the left translation of <math>s</math> by <math>R_{\hat e}</math>, <center><math>L_{R_{\hat e}}: [0,R_{\hat e}]\times[0,\pi] \to [-R_{\hat e},0] \times [0,\pi].</math></center><br />
** We define our glued surface <math>\#_{\underline r}(\underline D)_v</math> by using strip coordinates to identify neighborhoods of each pair of gluing edges in <math>\hat E^g_v</math> with <math>[0,\infty)\times [0,\pi]</math> (in the component closer to the root) resp. <math>(-\infty,0]\times[0,\pi]</math> (in the component further from the root), and then replacing these two ends by the quotient space <center><math>([0,R_{\hat e}]\times[0,\pi])\sqcup([-R_{\hat e},0]\times[0,\pi]) / L_{R_{\hat e}}.</math></center> The resulting Riemann surface <math>\#_{\underline r}(\underline D)_v</math> is biholomorphic to a disk, and all the information (boundary marked points, interior marked points, etc.) persists in the glued disk.<br />
* We perform a similar gluing construction at each of the interior nodes, using the families of cylinder coordinates we chose earlier.<br />
<br />
== The atlas of Deligne-Mumford space ==<br />
<br />
We define charts for <math>DM(k,m;(A_j))^{\text{ord}}</math> like so:<br />
* Fix an element <math>\sigma \in DM(k,m;(A_j))^{\text{ord}}</math> and a representative <math>\hat\mu = (\hat T,\hat{\underline\ell},\hat{\underline x},\hat{\underline o})</math> of <math>\sigma</math>.<br />
* Choose a local slice near <math>\hat\mu</math>:<br />
** for every disk component with <math>\geq 3</math> boundary marked points, fix the first three boundary marked points;<br />
** for every disk component with <math>\leq 2</math> boundary marked points, fix the distinguished boundary marked points and the first interior marked point;<br />
** for every sphere component, fix the first three marked points.<br />
* The restriction to this local slice of the gluing map <math>[\#]</math> defined above is our chart.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=588Deligne-Mumford space2017-06-10T20:39:26Z<p>Natebottman: /* The atlas of Deligne-Mumford space */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v})</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above). For any vertex <math>v \in \#_{\underline r}(\hat V)</math>, we denote by <math>\hat V^g_v\subset \hat V</math> the vertices in <math>\hat T</math> which are identified to form <math>v</math>. These vertices form a subtree of <math>\hat T</math>; denote the edges in this subtree by <math>\hat E^g_{\underline r}</math>.<br />
* To define the objects <math>\#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)</math>, we will "glue" certain disks and spheres together, using the choices of strip coordinates and cylinder coordinates made above. The crucial constructions will be, for each main vertex <math>v \in \#_{\underline r}(\hat V)</math>, a glued disk <math>\#_{\underline r}(\underline D)_v</math> formed by gluing together the disks corresponding to the vertices in <math>\hat E^g_v</math>. We now make a few definitions towards the construction of <math>\#_{\underline r}(\underline D)_v</math>:<br />
** For <math>\hat e \in \hat E^g_v</math>, define the '''gluing length''' <math>R_{\hat e} := R(r_{\hat e})</math>.<br />
** Denote by <math>L_{R_{\hat e}}</math> the left translation of <math>s</math> by <math>R_{\hat e}</math>, <center><math>L_{R_{\hat e}}: [0,R_{\hat e}]\times[0,\pi] \to [-R_{\hat e},0] \times [0,\pi].</math></center><br />
** We define our glued surface <math>\#_{\underline r}(\underline D)_v</math> by using strip coordinates to identify neighborhoods of each pair of gluing edges in <math>\hat E^g_v</math> with <math>[0,\infty)\times [0,\pi]</math> (in the component closer to the root) resp. <math>(-\infty,0]\times[0,\pi]</math> (in the component further from the root), and then replacing these two ends by the quotient space <center><math>([0,R_{\hat e}]\times[0,\pi])\sqcup([-R_{\hat e},0]\times[0,\pi]) / L_{R_{\hat e}}.</math></center> The resulting Riemann surface <math>\#_{\underline r}(\underline D)_v</math> is biholomorphic to a disk, and all the information (boundary marked points, interior marked points, etc.) persists in the glued disk.<br />
* We perform a similar gluing construction at each of the interior nodes, using the families of cylinder coordinates we chose earlier.<br />
<br />
== The atlas of Deligne-Mumford space ==<br />
<br />
We define charts for <math>DM(k,m;(A_j))^{\text{ord}}</math> like so:<br />
* Fix an element <math>\sigma \in DM(k,m;(A_j))^{\text{ord}}</math> and a representative <math>\hat\mu = (\hat T,\hat{\underline\ell},\hat{\underline x},\hat{\underline o})</math> of <math>\sigma</math>.<br />
* Choose a local slice near <math>\hat\mu</math>:<br />
** for every disk component with <math>\geq 3</math> boundary marked points, fix the first three boundary marked points;<br />
** for every disk component with <math>\leq 2</math> boundary marked points, fix the distinguished boundary marked points and the first interior marked point;<br />
** for every sphere component, fix the first three marked points.<br />
* The restriction to this local slice of the gluing map <math>[\#]</math> defined above is our chart.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Problems_on_Deligne-Mumford_spaces&diff=587Problems on Deligne-Mumford spaces2017-06-10T16:12:34Z<p>Natebottman: /* Down and dirty with low-dimensional associahedra */</p>
<hr />
<div>These problems deal with Deligne-Mumford spaces, by which we mean the moduli spaces of domains relevant for our construction of the Fukaya category.<br />
The first problem forms the '''warm-up portion''': you should make sure you understand how to do this one before Tuesday morning.<br />
The remaining two problems form the '''further fun section''': useful for deeper understanding, but not essential for following the thread of the lectures.<br />
<br />
== Down and dirty with low-dimensional associahedra ==<br />
<br />
Using the notation of [[Deligne-Mumford space]], for any <math>d\geq 2</math>, the '''associahedron''' <math>\overline\mathcal{M}_{d+1} := DM(d,0,(\{0\},\ldots,\{d\}))</math> is a <math>(d-2)</math>-dimensional manifold with boundary and corners which parametrizes nodal trees of disks with <math>d+1</math> marked points, one of them distinguished (we think of the <math>d</math> undistinguished resp. 1 distinguished marked points as "input" resp. "output" marked points).<br />
The associahedra are one of the Deligne-Mumford spaces we will use during the summer school, corresponding to the situation where all the Lagrangians labeling the boundary segments are distinct.<br />
<br />
'''(a)''' As shown in [Auroux, Ex. 2.6 [[https://arxiv.org/pdf/1301.7056.pdf]]], <math>\overline\mathcal{M}_4</math> is homeomorphic to a closed interval, with one endpoint corresponding to a collision of the first two inputs <math>z_1, z_2</math> and the other corresponding to a collision of <math>z_2, z_3</math>.<br />
Moving up a dimension, <math>\overline\mathcal{M}_5</math> is a pentagon; it can be identified with the central pentagon in the depiction of <math>DM(4,0;(\{0,1,2,3,4\}))</math> in [[Deligne-Mumford space]].<br />
Which polyhedron is <math>\overline\mathcal{M}_6</math> equal to?<br />
(A good way to get started on this problem is to list the codimension-1 strata.)<br />
<br />
'''(b)''' Using the manifold-with-corners structure of the associahedra constructed in [[Deligne-Mumford space]], observe that the <math>1</math>-dimensional associahedron <math>\overline\mathcal{M}_4</math> can be covered by three charts:<br />
* boundary charts centered respectively at the two points in <math>\overline\mathcal{M}_4 \setminus \mathcal{M}_4</math> (the domains of these charts are of the form <math>[0,a)</math>, and a choice of a point in this interval tells us how much to smooth the node);<br />
* an interior chart (which we produce by fixing the positions of three of the marked points, and varying the positive of the fourth).<br />
Explicitly work out these charts, and the transition maps amongst them.<br />
<br />
== The poset indexing the strata of the associahedra ==<br />
<br />
The associahedron <math>\overline\mathcal{M}_{d+1}</math> can be given the structure of a stratified space, where the underlying poset is called <math>K_d</math> and consists of '''stable rooted ribbon trees with <math>d</math> leaves'''.<br />
Similarly to the setup in [[Moduli spaces of pseudoholomorphic polygons]], a stable rooted ribbon tree is a tree <math>T</math> satisfying these properties:<br />
* <math>T</math> has <math>d</math> leaves and 1 root (in our terminology, the root has valence 1 but is not counted as a leaf);<br />
* <math>T</math> is stable, i.e. every main vertex (=neither a leaf nor the root) has valence at least 3;<br />
* <math>T</math> is a ribbon tree, i.e. the edges incident to any vertex are equipped with a cyclic ordering.<br />
To define the partial order, we declare <math>T' \leq T</math> if we can contract some of the interior edges in <math>T'</math> to get <math>T</math>; we declare that <math>T'</math> is in the closure of <math>T</math> if <math>T'\leq T</math>.<br />
Write the closure of the stratum corresponding to <math>T</math> as a product of lower-dimensional <math>K_d</math>'s.<br />
Which tree corresponds to the top stratum of <math>\overline\mathcal{M}_{d+1}</math>?<br />
To the codimension-1 strata of <math>\overline\mathcal{M}_{d+1}</math>?<br />
<br />
...and, to the operadically initiated (or willing to dig around a little at [[http://ncatlab.org]]): show that the collection <math>(K_d)_{d\geq 2}</math> can be given the structure of an '''operad''' (which is to say that for every <math>d, e \geq 2</math> and <math>1 \leq i \leq d</math> there is a composition operation <math>\circ_i\colon K_d \times K_e \to K_{d+e-1}</math> which splices <math>T_e \in K_e</math> onto <math>T_d \in K_d</math> by identifying the outgoing edge of <math>T_e</math> with the <math>i</math>-th incoming edge of <math>T_d</math>, and that these operations satisfy some coherence conditions).<br />
Next, show that algebras / categories over the operad <math>(C_*(K_d))_{d\geq2}</math> of cellular chains on <math>K_d</math> are the same thing as <math>A_\infty</math> algebra / categories.<br />
<br />
== More low-dimensional examples of Deligne-Mumford spaces ==<br />
<br />
'''(a)''' Explicitly work out the 3-dimensional Deligne-Mumford space <math>DM(3,1;(\{0\},\{1\},\{2\},\{3\}))</math>.<br />
<br />
'''(b)''' For <math>d \geq 2</math> define <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C}) := DM(d)^{\text{sph,ord}}</math>.<br />
That is, if we define<br />
<center><math><br />
\mathcal{M}_{0,d+1}(\mathbb{C}) := \bigl\{ \underline{z} = \{z_0,\ldots,z_d\}\in \mathbb{CP}^1 \;\text{pairwise disjoint} \bigr\}/_\sim,<br />
</math></center><br />
where two configurations are identified if one can be taken to the other by a Moebius transformation, then <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C})</math> is the compactification defined by including stable trees of spheres, where every sphere has at least 3 special (marked/nodal) points and any neighboring pair of spheres is attached at a pair of points.<br />
(A detailed construction of <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C})</math> can be found in [big McDuff-Salamon, App. D].)<br />
Make the identifications (don't worry too much about rigor) <math>\overline\mathcal{M}_{0,4}(\mathbb{C}) \cong \mathrm{pt}</math>, <math>\overline\mathcal{M}_{0,5}(\mathbb{C}) \cong \mathbb{CP}^1</math>, and <math>\overline\mathcal{M}_{0,6}(\mathbb{C}) \cong (\mathbb{CP}^1\times\mathbb{CP}^1) \# 3\overline{\mathbb{CP}}^2</math>.</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Problems_on_Deligne-Mumford_spaces&diff=586Problems on Deligne-Mumford spaces2017-06-10T16:11:43Z<p>Natebottman: </p>
<hr />
<div>These problems deal with Deligne-Mumford spaces, by which we mean the moduli spaces of domains relevant for our construction of the Fukaya category.<br />
The first problem forms the '''warm-up portion''': you should make sure you understand how to do this one before Tuesday morning.<br />
The remaining two problems form the '''further fun section''': useful for deeper understanding, but not essential for following the thread of the lectures.<br />
<br />
== Down and dirty with low-dimensional associahedra ==<br />
<br />
Using the notation of [[Deligne-Mumford space]], for any <math>d\geq 2</math>, the '''associahedron''' <math>\overline\mathcal{M}_{d+1} := DM(d,0,(\{0\},\ldots,\{d\}))</math> is a <math>(d-2)</math>-dimensional manifold with boundary and corners which parametrizes nodal trees of disks with <math>d+1</math> marked points, one of them distinguished (we think of the <math>d</math> undistinguished resp. 1 distinguished marked points as "input" resp. "output" marked points).<br />
The associahedra are one of the Deligne-Mumford spaces we will use during the summer school, corresponding to the situation where all the Lagrangians labeling the boundary segments are distinct.<br />
<br />
'''(a)''' As shown in [Auroux, Ex. 2.6 [[https://arxiv.org/pdf/1301.7056.pdf]]], <math>\overline\mathcal{M}_4</math> is homeomorphic to a closed interval, with one endpoint corresponding to a collision of the first two inputs <math>z_1, z_2</math> and the other corresponding to a collision of <math>z_2, z_3</math>.<br />
Moving up a dimension, <math>\overline\mathcal{M}_5</math> is a pentagon; it can be identified with the central pentagon in the depiction of <math>DM(4,0;(\{0,1,2,3,4\}))</math> in [[Deligne-Mumford space]].<br />
Which polyhedron is <math>\overline\mathcal{M}_6</math> equal to?<br />
(A good way to get started on this problem is to list the codimension-1 strata.)<br />
<br />
'''(b)''' Using the manifold-with-corners structure of the associahedra constructed in [[Deligne-Mumford space]], observe that <math>\overline\mathcal{M}_3</math> can be covered by three charts:<br />
* boundary charts centered respectively at the two points in <math>\overline\mathcal{M}_3 \setminus \mathcal{M}_3</math> (the domains of these charts are of the form <math>[0,a)</math>, and a choice of a point in this interval tells us how much to smooth the node);<br />
* an interior chart (which we produce by fixing the positions of three of the marked points, and varying the positive of the fourth).<br />
Explicitly work out these charts, and the transition maps amongst them.<br />
<br />
== The poset indexing the strata of the associahedra ==<br />
<br />
The associahedron <math>\overline\mathcal{M}_{d+1}</math> can be given the structure of a stratified space, where the underlying poset is called <math>K_d</math> and consists of '''stable rooted ribbon trees with <math>d</math> leaves'''.<br />
Similarly to the setup in [[Moduli spaces of pseudoholomorphic polygons]], a stable rooted ribbon tree is a tree <math>T</math> satisfying these properties:<br />
* <math>T</math> has <math>d</math> leaves and 1 root (in our terminology, the root has valence 1 but is not counted as a leaf);<br />
* <math>T</math> is stable, i.e. every main vertex (=neither a leaf nor the root) has valence at least 3;<br />
* <math>T</math> is a ribbon tree, i.e. the edges incident to any vertex are equipped with a cyclic ordering.<br />
To define the partial order, we declare <math>T' \leq T</math> if we can contract some of the interior edges in <math>T'</math> to get <math>T</math>; we declare that <math>T'</math> is in the closure of <math>T</math> if <math>T'\leq T</math>.<br />
Write the closure of the stratum corresponding to <math>T</math> as a product of lower-dimensional <math>K_d</math>'s.<br />
Which tree corresponds to the top stratum of <math>\overline\mathcal{M}_{d+1}</math>?<br />
To the codimension-1 strata of <math>\overline\mathcal{M}_{d+1}</math>?<br />
<br />
...and, to the operadically initiated (or willing to dig around a little at [[http://ncatlab.org]]): show that the collection <math>(K_d)_{d\geq 2}</math> can be given the structure of an '''operad''' (which is to say that for every <math>d, e \geq 2</math> and <math>1 \leq i \leq d</math> there is a composition operation <math>\circ_i\colon K_d \times K_e \to K_{d+e-1}</math> which splices <math>T_e \in K_e</math> onto <math>T_d \in K_d</math> by identifying the outgoing edge of <math>T_e</math> with the <math>i</math>-th incoming edge of <math>T_d</math>, and that these operations satisfy some coherence conditions).<br />
Next, show that algebras / categories over the operad <math>(C_*(K_d))_{d\geq2}</math> of cellular chains on <math>K_d</math> are the same thing as <math>A_\infty</math> algebra / categories.<br />
<br />
== More low-dimensional examples of Deligne-Mumford spaces ==<br />
<br />
'''(a)''' Explicitly work out the 3-dimensional Deligne-Mumford space <math>DM(3,1;(\{0\},\{1\},\{2\},\{3\}))</math>.<br />
<br />
'''(b)''' For <math>d \geq 2</math> define <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C}) := DM(d)^{\text{sph,ord}}</math>.<br />
That is, if we define<br />
<center><math><br />
\mathcal{M}_{0,d+1}(\mathbb{C}) := \bigl\{ \underline{z} = \{z_0,\ldots,z_d\}\in \mathbb{CP}^1 \;\text{pairwise disjoint} \bigr\}/_\sim,<br />
</math></center><br />
where two configurations are identified if one can be taken to the other by a Moebius transformation, then <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C})</math> is the compactification defined by including stable trees of spheres, where every sphere has at least 3 special (marked/nodal) points and any neighboring pair of spheres is attached at a pair of points.<br />
(A detailed construction of <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C})</math> can be found in [big McDuff-Salamon, App. D].)<br />
Make the identifications (don't worry too much about rigor) <math>\overline\mathcal{M}_{0,4}(\mathbb{C}) \cong \mathrm{pt}</math>, <math>\overline\mathcal{M}_{0,5}(\mathbb{C}) \cong \mathbb{CP}^1</math>, and <math>\overline\mathcal{M}_{0,6}(\mathbb{C}) \cong (\mathbb{CP}^1\times\mathbb{CP}^1) \# 3\overline{\mathbb{CP}}^2</math>.</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=585Deligne-Mumford space2017-06-10T15:30:55Z<p>Natebottman: /* The atlas of Deligne-Mumford space */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v})</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above). For any vertex <math>v \in \#_{\underline r}(\hat V)</math>, we denote by <math>\hat V^g_v\subset \hat V</math> the vertices in <math>\hat T</math> which are identified to form <math>v</math>. These vertices form a subtree of <math>\hat T</math>; denote the edges in this subtree by <math>\hat E^g_{\underline r}</math>.<br />
* To define the objects <math>\#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)</math>, we will "glue" certain disks and spheres together, using the choices of strip coordinates and cylinder coordinates made above. The crucial constructions will be, for each main vertex <math>v \in \#_{\underline r}(\hat V)</math>, a glued disk <math>\#_{\underline r}(\underline D)_v</math> formed by gluing together the disks corresponding to the vertices in <math>\hat E^g_v</math>. We now make a few definitions towards the construction of <math>\#_{\underline r}(\underline D)_v</math>:<br />
** For <math>\hat e \in \hat E^g_v</math>, define the '''gluing length''' <math>R_{\hat e} := R(r_{\hat e})</math>.<br />
** Denote by <math>L_{R_{\hat e}}</math> the left translation of <math>s</math> by <math>R_{\hat e}</math>, <center><math>L_{R_{\hat e}}: [0,R_{\hat e}]\times[0,\pi] \to [-R_{\hat e},0] \times [0,\pi].</math></center><br />
** We define our glued surface <math>\#_{\underline r}(\underline D)_v</math> by using strip coordinates to identify neighborhoods of each pair of gluing edges in <math>\hat E^g_v</math> with <math>[0,\infty)\times [0,\pi]</math> (in the component closer to the root) resp. <math>(-\infty,0]\times[0,\pi]</math> (in the component further from the root), and then replacing these two ends by the quotient space <center><math>([0,R_{\hat e}]\times[0,\pi])\sqcup([-R_{\hat e},0]\times[0,\pi]) / L_{R_{\hat e}}.</math></center> The resulting Riemann surface <math>\#_{\underline r}(\underline D)_v</math> is biholomorphic to a disk, and all the information (boundary marked points, interior marked points, etc.) persists in the glued disk.<br />
* We perform a similar gluing construction at each of the interior nodes, using the families of cylinder coordinates we chose earlier.<br />
<br />
== The atlas of Deligne-Mumford space ==<br />
<br />
We will define a chart for <math>DM(k,m;(A_j))^{\text{ord}}</math> like so:<br />
* Fix an element <math>\sigma \in DM(k,m;(A_j))^{\text{ord}}</math> and a representative <math>\hat\mu = (\hat T,\hat{\underline\ell},\hat{\underline x},\hat{\underline o})</math> of <math>\sigma</math>.<br />
* Choose a local slice near <math>\hat\mu</math>:<br />
** for every disk component with <math>\geq 3</math> boundary marked points, fix the first three boundary marked points;<br />
** for every disk component with <math>\leq 2</math> boundary marked points, fix the distinguished boundary marked points and the first interior marked point;<br />
** for every sphere component, fix the first three marked points.<br />
* The restriction to this local slice of the gluing map <math>[\#]</math> defined above is our chart.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=584Deligne-Mumford space2017-06-10T15:02:05Z<p>Natebottman: </p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v})</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above). For any vertex <math>v \in \#_{\underline r}(\hat V)</math>, we denote by <math>\hat V^g_v\subset \hat V</math> the vertices in <math>\hat T</math> which are identified to form <math>v</math>. These vertices form a subtree of <math>\hat T</math>; denote the edges in this subtree by <math>\hat E^g_{\underline r}</math>.<br />
* To define the objects <math>\#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)</math>, we will "glue" certain disks and spheres together, using the choices of strip coordinates and cylinder coordinates made above. The crucial constructions will be, for each main vertex <math>v \in \#_{\underline r}(\hat V)</math>, a glued disk <math>\#_{\underline r}(\underline D)_v</math> formed by gluing together the disks corresponding to the vertices in <math>\hat E^g_v</math>. We now make a few definitions towards the construction of <math>\#_{\underline r}(\underline D)_v</math>:<br />
** For <math>\hat e \in \hat E^g_v</math>, define the '''gluing length''' <math>R_{\hat e} := R(r_{\hat e})</math>.<br />
** Denote by <math>L_{R_{\hat e}}</math> the left translation of <math>s</math> by <math>R_{\hat e}</math>, <center><math>L_{R_{\hat e}}: [0,R_{\hat e}]\times[0,\pi] \to [-R_{\hat e},0] \times [0,\pi].</math></center><br />
** We define our glued surface <math>\#_{\underline r}(\underline D)_v</math> by using strip coordinates to identify neighborhoods of each pair of gluing edges in <math>\hat E^g_v</math> with <math>[0,\infty)\times [0,\pi]</math> (in the component closer to the root) resp. <math>(-\infty,0]\times[0,\pi]</math> (in the component further from the root), and then replacing these two ends by the quotient space <center><math>([0,R_{\hat e}]\times[0,\pi])\sqcup([-R_{\hat e},0]\times[0,\pi]) / L_{R_{\hat e}}.</math></center> The resulting Riemann surface <math>\#_{\underline r}(\underline D)_v</math> is biholomorphic to a disk, and all the information (boundary marked points, interior marked points, etc.) persists in the glued disk.<br />
* We perform a similar gluing construction at each of the interior nodes, using the families of cylinder coordinates we chose earlier.<br />
<br />
== The atlas of Deligne-Mumford space ==<br />
<br />
We will define a chart for <math>DM(k,m;(A_j))</math><br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=571Deligne-Mumford space2017-06-09T20:20:04Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v})</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above). For any vertex <math>v \in \#_{\underline r}(\hat V)</math>, we denote by <math>\hat V^g_v\subset \hat V</math> the vertices in <math>\hat T</math> which are identified to form <math>v</math>. These vertices form a subtree of <math>\hat T</math>; denote the edges in this subtree by <math>\hat E^g_{\underline r}</math>.<br />
* To define the objects <math>\#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)</math>, we will "glue" certain disks and spheres together, using the choices of strip coordinates and cylinder coordinates made above. The crucial constructions will be, for each main vertex <math>v \in \#_{\underline r}(\hat V)</math>, a glued disk <math>\#_{\underline r}(\underline D)_v</math> formed by gluing together the disks corresponding to the vertices in <math>\hat E^g_v</math>. We now make a few definitions towards the construction of <math>\#_{\underline r}(\underline D)_v</math>:<br />
** For <math>\hat e \in \hat E^g_v</math>, define the '''gluing length''' <math>R_{\hat e} := R(r_{\hat e})</math>.<br />
** Denote by <math>L_{R_{\hat e}}</math> the left translation of <math>s</math> by <math>R_{\hat e}</math>, <center><math>L_{R_{\hat e}}: [0,R_{\hat e}]\times[0,\pi] \to [-R_{\hat e},0] \times [0,\pi].</math></center><br />
** We define our glued surface <math>\#_{\underline r}(\underline D)_v</math> by using strip coordinates to identify neighborhoods of each pair of gluing edges in <math>\hat E^g_v</math> with <math>[0,\infty)\times [0,\pi]</math> (in the component closer to the root) resp. <math>(-\infty,0]\times[0,\pi]</math> (in the component further from the root), and then replacing these two ends by the quotient space <center><math>([0,R_{\hat e}]\times[0,\pi])\sqcup([-R_{\hat e},0]\times[0,\pi]) / L_{R_{\hat e}}.</math></center> The resulting Riemann surface <math>\#_{\underline r}(\underline D)_v</math> is biholomorphic to a disk, and all the information (boundary marked points, interior marked points, etc.) persists in the glued disk.<br />
* We perform a similar gluing construction at each of the interior nodes, using the families of cylinder coordinates we chose earlier.<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=570Deligne-Mumford space2017-06-09T20:19:17Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}} \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v})</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above). For any vertex <math>v \in \#_{\underline r}(\hat V)</math>, we denote by <math>\hat V^g_v\subset \hat V</math> the vertices in <math>\hat T</math> which are identified to form <math>v</math>. These vertices form a subtree of <math>\hat T</math>; denote the edges in this subtree by <math>\hat E^g_{\underline r}</math>.<br />
* To define the objects <math>\#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)</math>, we will "glue" certain disks and spheres together, using the choices of strip coordinates and cylinder coordinates made above. The crucial constructions will be, for each main vertex <math>v \in \#_{\underline r}(\hat V)</math>, a glued disk <math>\#_{\underline r}(\underline D)_v</math> formed by gluing together the disks corresponding to the vertices in <math>\hat E^g_v</math>. We now make a few definitions towards the construction of <math>\#_{\underline r}(\underline D)_v</math>:<br />
** For <math>\hat e \in \hat E^g_v</math>, define the '''gluing length''' <math>R_{\hat e} := R(r_{\hat e})</math>.<br />
** Denote by <math>L_{R_{\hat e}}</math> the left translation of <math>s</math> by <math>R_{\hat e}</math>, <center><math>L_{R_{\hat e}}: [0,R_{\hat e}]\times[0,\pi] \to [-R_{\hat e},0] \times [0,\pi].</math></center><br />
** We define our glued surface <math>\#_{\underline r}(\underline D)_v</math> by using strip coordinates to identify neighborhoods of each pair of gluing edges in <math>\hat E^g_v</math> with <math>[0,\infty)\times [0,\pi]</math> (in the component closer to the root) resp. <math>(-\infty,0]\times[0,\pi]</math> (in the component further from the root), and then replacing these two ends by the quotient space <center><math>([0,R_{\hat e}]\times[0,\pi])\sqcup([-R_{\hat e},0]\times[0,\pi]) / L_{R_{\hat e}}.</math></center> The resulting Riemann surface <math>\#_{\underline r}(\underline D)_v</math> is biholomorphic to a disk, and all the information (boundary marked points, interior marked points, etc.) persists in the glued disk.<br />
* We perform a similar gluing construction at each of the interior nodes, using the families of cylinder coordinates we chose earlier.<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=569Deligne-Mumford space2017-06-09T20:17:10Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}} \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v})</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above). For any vertex <math>v \in \#_{\underline r}(\hat V)</math>, we denote by <math>\hat V^g_v\subset \hat V</math> the vertices in <math>\hat T</math> which are identified to form <math>v</math>. These vertices form a subtree of <math>\hat T</math>; denote the edges in this subtree by <math>\hat E^g_{\underline r}</math>.<br />
* To define the objects <math>\#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)</math>, we will "glue" certain disks and spheres together, using the choices of strip coordinates and cylinder coordinates made above. The crucial constructions will be, for each main vertex <math>v \in \#_{\underline r}(\hat V)</math>, a glued disk <math>\#_{\underline r}(\underline D)_v</math> formed by gluing together the disks corresponding to the vertices in <math>\hat E^g_v</math>. We now make a few definitions towards the construction of <math>\#_{\underline r}(\underline D)_v</math>:<br />
** For <math>\hat e \in \hat E^g_v</math>, define the '''gluing length''' <math>R_{\hat e} := R(r_{\hat e})</math>.<br />
** Denote by <math>L_{R_{\hat e}}</math> the left translation of <math>s</math> by <math>R_{\hat e}</math>, <center><math>L_{R_{\hat e}}: [0,R_{\hat e}]\times[0,\pi] \to [-R_{\hat e},0] \times [0,\pi].</math></center><br />
** We define our glued surface <math>\#_{\underline r}(\underline D)_v</math> by using strip coordinates to identify neighborhoods of each pair of gluing edges in <math>\hat E^g_v</math> with <math>[0,\infty)\times [0,\pi]</math> (in the component closer to the root) resp. <math>(-\infty,0]\times[0,\pi]</math> (in the component further from the root), and then replacing these two ends by the quotient space <center><math>([0,R_{\hat e}]\times[0,\pi])\sqcup([-R_{\hat e},0]\times[0,\pi]) / L_{R_{\hat e}}.</math></center><br />
The resulting Riemann surface <math>\#_{\underline r}(\underline D)_v</math> is biholomorphic to a disk, and all the information (boundary marked points, interior marked points, etc.) persists in the glued disk.<br />
* We perform a similar gluing construction at each of the interior nodes, using the families of cylinder coordinates we chose earlier.<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Table_of_contents&diff=568Table of contents2017-06-09T20:02:57Z<p>Natebottman: </p>
<hr />
<div><br />
'''Table of contents''' for [[Polyfold Constructions for Fukaya Categories]]<br />
<br />
<br />
'''Construction overviews:'''<br />
<br />
* [[Moduli spaces of pseudoholomorphic polygons]]<br />
* [[Regularized moduli spaces]] <br />
* summary of resulting [[Floer chain complex]] (TODO)<br />
<br />
<br />
'''Algebra details:'''<br />
<br />
* [[Novikov ring]] (TODO)<br />
* [[Brane structure]] (TODO)<br />
** [[Grading induced by brane structures]] (TODO)<br />
* [[Orientations of Cauchy-Riemann sections]] induced by brane structures (TODO)<br />
<br />
<br />
'''Geometry/Topology/Combinatorics details:'''<br />
<br />
* [[Deligne-Mumford space]] (N-work-in-progress)<br />
** [[Deligne-Mumford space#Gluing and the definition of the topology|glued surface]] (N-work in progress -- and need to synchronize with reference to this in [[http://www.polyfolds.org/index.php?title=Moduli_spaces_of_pseudoholomorphic_polygons#General_moduli_space_of_pseudoholomorphic_polygons expansion of expansion in point 7]] and [[Gluing construction for Hamiltonians]] )<br />
* [[Compactified Morse trajectory spaces]] (brief summary with references - could use extension)<br />
* [[Coherent orientations on the regularized moduli spaces]] arising from polyfold theory (TODO)<br />
* the polyfold [[ambient space]] <math>\mathcal{X} (\underline{x}) </math> as a topological space (K-TODO)<br />
** [[Gromov topology]] (K-TODO)<br />
* the polyfold [[ambient bundle]] <math>\pi: \mathcal{Y}_J(\underline{x}) </math> as continuous surjection between topological spaces (K-TODO)<br />
* the [[Cauchy-Riemann section]] as continuous map <math>\overline\partial_{J,Y}:\mathcal{X}(\underline{x})\to \mathcal{Y}_J(\underline{x})</math> (K-TODO)<br />
** [[Gluing construction for Hamiltonians]] (TODO)<br />
<br />
<br />
'''Analysis details:'''<br />
<br />
* the [[polyfold smooth structure]] on the [[ambient space]] <math>\mathcal{X} (\underline{x}) </math> (J-TODO)<br />
* the [[polyfold bundle structure]] of the [[ambient bundle]] <math>\pi: \mathcal{Y}_J(\underline{x}) </math> (J-TODO)<br />
* the [[polyfold Fredholm property]] of the [[Cauchy-Riemann section]] <math>\overline\partial_{J,Y}:\mathcal{X}(\underline{x})\to \mathcal{Y}_J(\underline{x})</math> (J-TODO)<br />
* proof that [[Gromov compactness implies properness]] of the [[Cauchy-Riemann section]] (TODO)</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Table_of_contents&diff=567Table of contents2017-06-09T20:01:43Z<p>Natebottman: </p>
<hr />
<div><br />
'''Table of contents''' for [[Polyfold Constructions for Fukaya Categories]]<br />
<br />
<br />
'''Construction overviews:'''<br />
<br />
* [[Moduli spaces of pseudoholomorphic polygons]]<br />
* [[Regularized moduli spaces]] <br />
* summary of resulting [[Floer chain complex]] (TODO)<br />
<br />
<br />
'''Algebra details:'''<br />
<br />
* [[Novikov ring]] (TODO)<br />
* [[Brane structure]] (TODO)<br />
** [[Grading induced by brane structures]] (TODO)<br />
* [[Orientations of Cauchy-Riemann sections]] induced by brane structures (TODO)<br />
<br />
<br />
'''Geometry/Topology/Combinatorics details:'''<br />
<br />
* [[Deligne-Mumford space]] (Nate-work-in-progress)<br />
** [[Deligne-Mumford space#Gluing and the definition of the topology|glued surface]] (N-TODO .. or replace reference to this in [[http://www.polyfolds.org/index.php?title=Moduli_spaces_of_pseudoholomorphic_polygons#General_moduli_space_of_pseudoholomorphic_polygons expansion of expansion in point 7]] and [[Gluing construction for Hamiltonians]] )<br />
* [[Compactified Morse trajectory spaces]] (brief summary with references - could use extension)<br />
* [[Coherent orientations on the regularized moduli spaces]] arising from polyfold theory (TODO)<br />
* the polyfold [[ambient space]] <math>\mathcal{X} (\underline{x}) </math> as a topological space (K-TODO)<br />
** [[Gromov topology]] (K-TODO)<br />
* the polyfold [[ambient bundle]] <math>\pi: \mathcal{Y}_J(\underline{x}) </math> as continuous surjection between topological spaces (K-TODO)<br />
* the [[Cauchy-Riemann section]] as continuous map <math>\overline\partial_{J,Y}:\mathcal{X}(\underline{x})\to \mathcal{Y}_J(\underline{x})</math> (K-TODO)<br />
** [[Gluing construction for Hamiltonians]] (TODO)<br />
<br />
<br />
'''Analysis details:'''<br />
<br />
* the [[polyfold smooth structure]] on the [[ambient space]] <math>\mathcal{X} (\underline{x}) </math> (J-TODO)<br />
* the [[polyfold bundle structure]] of the [[ambient bundle]] <math>\pi: \mathcal{Y}_J(\underline{x}) </math> (J-TODO)<br />
* the [[polyfold Fredholm property]] of the [[Cauchy-Riemann section]] <math>\overline\partial_{J,Y}:\mathcal{X}(\underline{x})\to \mathcal{Y}_J(\underline{x})</math> (J-TODO)<br />
* proof that [[Gromov compactness implies properness]] of the [[Cauchy-Riemann section]] (TODO)</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Table_of_contents&diff=566Table of contents2017-06-09T20:00:44Z<p>Natebottman: </p>
<hr />
<div><br />
'''Table of contents''' for [[Polyfold Constructions for Fukaya Categories]]<br />
<br />
<br />
'''Construction overviews:'''<br />
<br />
* [[Moduli spaces of pseudoholomorphic polygons]]<br />
* [[Regularized moduli spaces]] <br />
* summary of resulting [[Floer chain complex]] (TODO)<br />
<br />
<br />
'''Algebra details:'''<br />
<br />
* [[Novikov ring]] (TODO)<br />
* [[Brane structure]] (TODO)<br />
** [[Grading induced by brane structures]] (TODO)<br />
* [[Orientations of Cauchy-Riemann sections]] induced by brane structures (TODO)<br />
<br />
<br />
'''Geometry/Topology/Combinatorics details:'''<br />
<br />
* [[Deligne-Mumford space]] (Nate-work-in-progress)<br />
** [[Deligne-Mumford space#Gluing and the definition of the topology]] (N-TODO .. or replace reference to this in [[http://www.polyfolds.org/index.php?title=Moduli_spaces_of_pseudoholomorphic_polygons#General_moduli_space_of_pseudoholomorphic_polygons expansion of expansion in point 7]] and [[Gluing construction for Hamiltonians]] )<br />
* [[Compactified Morse trajectory spaces]] (brief summary with references - could use extension)<br />
* [[Coherent orientations on the regularized moduli spaces]] arising from polyfold theory (TODO)<br />
* the polyfold [[ambient space]] <math>\mathcal{X} (\underline{x}) </math> as a topological space (K-TODO)<br />
** [[Gromov topology]] (K-TODO)<br />
* the polyfold [[ambient bundle]] <math>\pi: \mathcal{Y}_J(\underline{x}) </math> as continuous surjection between topological spaces (K-TODO)<br />
* the [[Cauchy-Riemann section]] as continuous map <math>\overline\partial_{J,Y}:\mathcal{X}(\underline{x})\to \mathcal{Y}_J(\underline{x})</math> (K-TODO)<br />
** [[Gluing construction for Hamiltonians]] (TODO)<br />
<br />
<br />
'''Analysis details:'''<br />
<br />
* the [[polyfold smooth structure]] on the [[ambient space]] <math>\mathcal{X} (\underline{x}) </math> (J-TODO)<br />
* the [[polyfold bundle structure]] of the [[ambient bundle]] <math>\pi: \mathcal{Y}_J(\underline{x}) </math> (J-TODO)<br />
* the [[polyfold Fredholm property]] of the [[Cauchy-Riemann section]] <math>\overline\partial_{J,Y}:\mathcal{X}(\underline{x})\to \mathcal{Y}_J(\underline{x})</math> (J-TODO)<br />
* proof that [[Gromov compactness implies properness]] of the [[Cauchy-Riemann section]] (TODO)</div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=564Deligne-Mumford space2017-06-09T19:45:51Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}} \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v})</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above). For any vertex <math>v \in \#_{\underline r}(\hat V)</math>, we denote by <math>\hat V^g_v\subset \hat V</math> the vertices in <math>\hat T</math> which are identified to form <math>v</math>.<br />
* To define the objects <math>\#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)</math>, we will "glue" certain disks and spheres together, using the choices of strip coordinates and cylinder coordinates made above. The crucial constructions will be, for each main vertex <math>v \in \#_{\underline r}(\hat V)</math>, a glued disk <math>\#_{\underline r}(\underline D)_v</math> formed by gluing together the disks corresponding to the<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=563Deligne-Mumford space2017-06-09T15:40:29Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}} \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v}</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above).<br />
* To define the objects <math>\#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)</math>, we will "glue" certain disks and spheres together, using the choices of strip coordinates and cylinder coordinates made above. The crucial constructions will be, for each main vertex <math>v \in \#_{\underline r}(\hat V)</math><br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=562Deligne-Mumford space2017-06-09T15:27:01Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}} \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v}</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above).<br />
* To define the<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=561Deligne-Mumford space2017-06-09T14:33:36Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}} \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v}</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above).<br />
*<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=560Deligne-Mumford space2017-06-09T14:22:18Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}} \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\hat{\underline \ell}), \#_{\underline r}(\hat{\underline x}), \#_{\underline r}(\hat{\underline O}), \#_{\underline s}(\hat{\underline \beta}), \#_{\underline r}(\hat{\underline z})].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v}</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above).<br />
*<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=559Deligne-Mumford space2017-06-09T13:35:40Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}} \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\hat{\underline\ell},\hat{\underline x},\hat{\underline O},\hat{\underline\beta},\hat{\underline z})\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\hat{\underline \ell}), \#_{\underline r}(\hat{\underline x}), \#_{\underline r}(\hat{\underline O}), \#_{\underline s}(\hat{\underline \beta}), \#_{\underline r}(\hat{\underline z})].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''Floer-type nodal edge''' in <math>\hat T</math> ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>, "nodal" meaning that the edge connects two main vertices).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v}</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
We can now construct the gluing map <math>[\#]</math>.<br />
We will not include all the details.<br />
* The '''glued tree''' <math>\#_{\underline r}(\hat T)</math> is defined by collapsing the '''gluing edges''' <math>\hat E^g_{\underline r} := \{\hat e \in \hat E^{\text{nd}} \;|\; r_{\hat e}>0\}</math>, where the '''nodal edges''' <math>\hat E^{\text{nd}}</math> are the Floer-type nodal edges and Morse-type nodal edges (see above).<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=558Deligne-Mumford space2017-06-09T13:19:00Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in <math>\ell^\infty</math>-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}} \times \prod_{1\leq i\leq t} U_\epsilon^{\hat{\beta_i}}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\hat{\underline\ell},\hat{\underline x},\hat{\underline O},\hat{\underline\beta},\hat{\underline z})\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\hat{\underline \ell}), \#_{\underline r}(\hat{\underline x}), \#_{\underline r}(\hat{\underline O}), \#_{\underline s}(\hat{\underline \beta}), \#_{\underline r}(\hat{\underline z})].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''interior Floer-type edge''' in <math>\hat T</math> ("interior" meaning connecting two main vertices, "Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\hat\ell_{\hat e}=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\hat{\underline O}</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\hat\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\hat \beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.<br />
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v}</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v}</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).<br />
** Similar restrictions apply to <math>\underline\beta</math>.<br />
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=557Deligne-Mumford space2017-06-09T13:00:53Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in Hausdorff distance, and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}} \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''interior Floer-type edge''' in <math>\hat T</math> ("interior" meaning connecting two main vertices, "Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single <math>A_j</math>).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''Morse-type nodal edge''' in <math>\hat T</math> ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single <math>A_j</math>, "nodal" meaning <math>\ell_e=0</math>).<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\underline O</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:<br />
** for every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1}</math>.<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=556Deligne-Mumford space2017-06-09T11:09:26Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in Hausdorff distance, and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times U_\epsilon^{\text{int}} \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\epsilon)</math>, one for every '''interior Floer edge''' in <math>\hat T</math> ("interior" meaning connecting two main vertices, "Floer edge" meaning an edge whose adjacent regions are labeled by pair not contained in a single <math>A_j</math>).<br />
* <math>U_\epsilon^{\text{Morse}}(\underline 0)</math> is a product of intervals <math>(-\epsilon,\epsilon)</math>, one for every '''interior Morse edge''' in <math>\hat T</math>.<br />
<math>U_\epsilon^{\text{int}}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every point in <math>\underline O</math> to which a sphere tree is attached.<br />
* <math>U_\epsilon^{\beta_i}(\underline 0)</math> is a product of disks <math>D_\epsilon(0)</math>, one for every interior edge in the unordered tree underlying the sphere tree <math>\beta_i</math>.<br />
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=554Deligne-Mumford space2017-06-09T00:03:48Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in Hausdorff distance, and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{Floer}}(\underline 0) \times U_\epsilon^{\text{Morse}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{Floer}}(\underline 0),U_\epsilon^{\text{Morse}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
* <math>U_\epsilon^{\text{Floer}}(\underline 0)</math> is a product of intervals <math>[0,\eps)</math><br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=553Deligne-Mumford space2017-06-08T23:59:06Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in Hausdorff distance, and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{disk}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing <math>0</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
With these preparatory choices made, we can define the sets <math>U_\epsilon^{\text{disk}}(\underline 0), U_\epsilon^{\beta_i}(\underline 0), U_\epsilon(\hat\mu)</math> appearing in the definition of <math>[\#]</math>.<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=552Deligne-Mumford space2017-06-08T23:46:10Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in Hausdorff distance, and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{disk}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates. This results from the fact that we are working with the disk <math>D</math> itself, not a complex curve abstractly isomorphic to <math>D</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=551Deligne-Mumford space2017-06-08T23:42:55Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in Hausdorff distance, and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{disk}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
* For each main vertex <math>\hat v</math> and point in <math>\underline O_v</math>, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.)<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
* <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
* <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates. This results from the fact that we are working with the disk <math>D</math> itself, not a complex curve abstractly isomorphic to <math>D</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=550Deligne-Mumford space2017-06-08T23:40:31Z<p>Natebottman: </p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in Hausdorff distance, and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{disk}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
Here we define the notion of (positive resp. negative) strip coordinates.<br />
<div class="mw-collapsible-content"> <br />
Given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
** <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
** <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates. This results from the fact that we are working with the disk <math>D</math> itself, not a complex curve abstractly isomorphic to <math>D</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
</div></div><br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=549Deligne-Mumford space2017-06-08T23:39:23Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in Hausdorff distance, and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{disk}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math> (this notion is defined below).<br />
<br />
given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
** <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
** <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates. This results from the fact that we are working with the disk <math>D</math> itself, not a complex curve abstractly isomorphic to <math>D</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=548Deligne-Mumford space2017-06-08T23:38:03Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in Hausdorff distance, and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{disk}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math>. More specifically, given <math>x \in \partial D</math>, we call biholomorphisms <center><math>h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},</math></center><center><math>h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math></center> into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
** <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \Im z \geq 0\}</math> are defined by <center><math>p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);</math></center><br />
** <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\; \Im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
(Note that there is only a 2-dimensional space of choices of strip coordinates. This results from the fact that we are working with the disk <math>D</math> itself, not a complex curve abstractly isomorphic to <math>D</math>.)<br />
Now that we've defined strip coordinates, we say that for <math>\hat x \in \partial D</math> and <math>x</math> varying in <math>U_\epsilon(\hat x)</math>, a '''family of positive resp. negative strip coordinates near <math>\hat x</math>''' is a collection of functions <center><math>h^+(x,\cdot): \mathbb{R}^+\times[0,\pi] \to D,</math></center><center><math>h^-(x,\cdot): \mathbb{R}^-\times[0,\pi]</math></center> if it is of the form <math>h^\pm(x,\cdot) = f_x\circ p^\pm</math>, where <math>f_x</math> is a smooth family of Moebius transformations sending <math>\{z\;|\;\Im z\geq0\}\cup\{\infty\}</math> to the disk, such that <math>f_x(0)=x</math> and <math>f_x(\infty)</math> is independent of <math>x</math>.<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=547Deligne-Mumford space2017-06-08T23:23:06Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in Hausdorff distance, and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{disk}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
* For each main vertex <math>\hat v \in \hat V^m</math> and edge <math>\hat e \in \hat E(\hat v)</math>, we must choose a family of strip coordinates <math>h_{\hat e}^\pm</math> near <math>\hat x_{\hat v,\hat e}</math>. More specifically, given <math>x \in \partial D</math>, we call a biholomorphisms<br />
<center><br />
<math><br />
h^+: \mathbb{R}^+ \times [0,\pi] \to N(x) \setminus \{x\},<br />
</math><br />
</center><br />
<center><br />
<math><br />
h^-: \mathbb{R}^- \times [0,\pi] \to N(x) \setminus \{x\}<br />
</math><br />
</center><br />
into a closed neighborhood <math>N(x) \subset D</math> of <math>x</math> '''positive resp. negative strip coordinates near <math>x</math>''' if it is of the form <math>h^\pm = f \circ p^\pm</math>, where<br />
** <math>p^\pm : \mathbb{R}^\pm \times [0,\pi] \to \{0 < |z| \leq 1, \; \im z \geq 0\}</math> are defined by<br />
<center><br />
<math><br />
p^+(z) := -\exp(-z), \quad p^-(z) := \exp(z);<br />
</math><br />
</center><br />
** <math>f</math> is a Moebius transformation which maps the extended upper half plane <math>\{z \;|\: \im z \geq 0\} \cup \{\infty\}</math> to the <math>D</math> and sends <math>0 \mapsto x</math>.<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=546Deligne-Mumford space2017-06-08T23:04:33Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, varying the marked points by less than <math>\epsilon</math> in Hausdorff distance, and varying the lengths <math>\ell_e</math> by less than <math>\epsilon</math>.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{disk}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
The image of this map is defined to be <math>U_\epsilon(\sigma;\hat\mu)</math>.<br />
<br />
Before we can construct <math>[\#]</math>, we need to make some choices:<br />
* Fix a '''gluing profile''', i.e. a continuous decreasing function <math>R: (0,1] \to [0,\infty)</math> with <math>\lim_{r \to 0} R(r) = \infty</math>, <math>R(1) = 0</math>. (For instance, we could use the '''exponential gluing profile''' <math>R(r) := \exp(1/r) - e</math>.)<br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=545Deligne-Mumford space2017-06-08T22:58:49Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, and varying the marked points by less than <math>\epsilon</math> in Hausdorff distance.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{disk}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
</math><br />
</center><br />
<center><br />
<math><br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=544Deligne-Mumford space2017-06-08T22:57:59Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, and varying the marked points by less than <math>\epsilon</math> in Hausdorff distance.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text{disk}}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\beta_i}(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
\quad<br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=543Deligne-Mumford space2017-06-08T22:57:15Z<p>Natebottman: </p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, and varying the marked points by less than <math>\epsilon</math> in Hausdorff distance.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon^{\text disk}(\underline 0) \times \prod_{1\leq i\leq t} U_\epsilon^{\text \beta_i}(\underline 0} \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
\quad<br />
\bigl(\underline r, (\underline s_i), (\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)\bigr) \mapsto [\#_{\underline r}(\hat T), \#_{\underline r}(\underline \ell), \#_{\underline r}(\underline x), \#_{\underline r}(\underline O), \#_{\underline s}(\underline \beta), \#_{\underline r}(\underline z)].<br />
</math><br />
</center><br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottmanhttp://www.polyfolds.org/index.php?title=Deligne-Mumford_space&diff=542Deligne-Mumford space2017-06-08T22:49:02Z<p>Natebottman: /* Gluing and the definition of the topology */</p>
<hr />
<div>'''in progress!'''<br />
<br />
[[table of contents]]<br />
<br />
Here we describe the domain moduli spaces needed for the construction of the composition operations <math>\mu^d</math> described in [[Polyfold Constructions for Fukaya Categories#Composition Operations]]; we refer to these as '''Deligne-Mumford spaces'''.<br />
In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished).<br />
We tailor these spaces to our needs:<br />
* In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in <math>[0,1]</math>. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.<br />
* Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.<br />
These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces.<br />
We largely follow Chapters 5-8 of [LW [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf]]], with two notable departures:<br />
* In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.<br />
* Li-Wehrheim work with a ''single'' Lagrangian, hence all interior edges are labeled by a number in <math>[0,1]</math>. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.<br />
<br />
== Some notation and examples ==<br />
<br />
For <math>k \geq 1</math>, <math>\ell \geq 0</math> with <math>k + 2\ell \geq 2</math> and for a decomposition <math>\{0,1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_m</math>, we will denote by <math>DM(k,\ell;(A_j)_{1\leq j\leq m})</math> the Deligne-Mumford space of nodal disks with <math>k</math> boundary marked points and <math>\ell</math> unordered interior marked points and with numbers in <math>[0,1]</math> labeling certain nodes.<br />
(This anticipates the space of nodal disk maps where, for <math>i \in A_j, i' \in A_{j'}</math>, <math>L_i</math> and <math>L_{i'}</math> are the same Lagrangian if and only if <math>j = j'</math>.)<br />
It will also be useful to consider the analogous spaces <math>DM(k,\ell;(A_j))^{\text{ord}}</math> where the interior marked points are ordered.<br />
As we will see, <math>DM(k,\ell;(A_j))^{\text{ord}}</math> is a manifold with boundary and corners, while <math>DM(k,\ell;(A_j))</math> is an orbifold; moreover, we have isomorphisms<br />
<center><br />
<math><br />
DM(k,\ell;(A_j))^{\text{ord}}/S_\ell \quad\stackrel{\simeq}{\longrightarrow}\quad DM(k,\ell;(A_j)),<br />
</math><br />
</center><br />
which are defined by forgetting the ordering of the interior marked points.<br />
<br />
Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.<br />
<br />
Here is <math>DM(3,0;(\{0,2,3\},\{1\}))</math>.<br />
<br />
'''TO DO'''<br />
<br />
Here is <math>DM(4,0;(\{0,1,2,3,4\}))</math>, with the associahedron <math>DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))</math> appearing as the smaller pentagon.<br />
Note that the hollow boundary marked point is the "distinguished" one.<br />
<br />
''need to edit to insert incoming/outgoing Morse trajectories''<br />
[[File:K4_ext.png | 1000px]]<br />
<br />
And here are <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(0,2;(\{0\},\{1\},\{2\}))</math> (lower-left) and <math>DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}} \to DM(1,2;(\{0\},\{1\},\{2\}))</math>.<br />
<math>DM(0,2;(\{0\},\{1\},\{2\}))</math> and <math>DM(1,2;(\{0\},\{1\},\{2\}))</math> have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere.<br />
The red bits in <math>DM(0,2;(\{0\},\{1\},\{2\}))^{\text{ord}}, DM(1,2;(\{0\},\{1\},\{2\}))^{\text{ord}}</math> are there only to show the points with nontrivial isotropy with respect to the <math>\mathbf{Z}/2</math>-action which interchanges the labels of the interior marked points.<br />
<br />
[[File:eyes.png | 1000px]]<br />
<br />
== Definitions ==<br />
<br />
Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks.<br />
We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.<br />
<br />
An '''ordered tree''' <math>T</math> is a tree satisfying these conditions:<br />
* there is a designated '''root vertex''' <math>\text{rt}(T) \in T</math>, which has valence <math>|\text{rt}(T)|=1</math>; we orient <math>T</math> toward the root.<br />
* <math>T</math> is '''ordered''' in the sense that for every <math>v \in T</math>, the incoming edge set <math>E^{\text{in}}(v)</math> is equipped with an order.<br />
* The vertex set <math>V</math> is partitioned <math>V = V^m \sqcup V^c</math> into '''main vertices''' and '''critical vertices''', such that the root is a critical vertex.<br />
An '''unordered tree''' is defined similarly, but without the order on incoming edges at every vertex; a '''labeled unordered tree''' is an unordered tree with the additional datum of a labeling of the non-root critical vertices.<br />
For instance, here is an example of an ordered tree <math>T</math>, with order corresponding to left-to-right order on the page:<br />
<br />
'''INSERT EXAMPLE HERE'''<br />
<br />
We now define, for <math>m \geq 2</math>, the space <math>DM(m)^{\text{sph}}</math> resp. <math>DM(m)^{\text{sph,ord}}</math> of nodal trees of spheres with <math>m</math> incoming unordered resp. ordered marked points.<br />
(These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, <math>DM(2)^{\text{sph,ord}} = \text{pt}</math>, <math>DM(3)^{\text{sph,ord}} \cong \mathbb{CP}^1</math>, and <math>DM(4)^{\text{sph,ord}}</math> is diffeomorphic to the blowup of <math>\mathbb{CP}^1\times\mathbb{CP}^1</math> at 3 points on the diagonal.)<br />
<center><br />
<math><br />
DM(m)^{\text{sph}} := \{(T,\underline O) \;|\; \text{(1),(2),(3) satisfied}\} /_\sim ,<br />
</math><br />
</center><br />
where (1), (2), (3) are as follows:<br />
# <math>T</math> is an unordered tree, with critical vertices consisting of <math>m</math> leaves and the root.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>O_{v,e}</math> is a marked point in <math>S^2</math> satisfying the requirement that for <math>v</math> fixed, the points <math>(O_{v,e})_{e \in E(v)}</math> are distinct. We denote this (unordered) set of marked points by <math>\underline o_v</math>.<br />
# The tree <math>(T,\underline O)</math> satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math>, we have <math>\#E(v) \geq 3</math>.<br />
The equivalence relation <math>\sim</math> is defined by declaring that <math>(T,\underline O)</math>, <math>(T',\underline O')</math> are '''equivalent''' if there exists an isomorphism of labeled trees <math>\zeta: T \to T'</math> and a collection <math>\underline\psi = (\psi_v)_{v \in V^m}</math> of holomorphism automorphisms of the sphere such that <math>\psi_v(\underline O_v) = \underline O_{\zeta(v)}'</math> for every <math>v \in V^m</math>.<br />
<br />
The spaces <math>DM(m)^{\text{sph,ord}}</math> are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.<br />
<br />
Next, we define, for <math>k \geq 0, m \geq 1, k+2m \geq 2</math> and a decomposition <math>\{1,\ldots,k\} = A_1 \sqcup \cdots \sqcup A_n</math>, the space <math>DM(k,m;(A_j))</math> of nodal trees of disks and spheres with <math>k</math> ordered incoming boundary marked points and <math>m</math> unordered interior marked points.<br />
Before we define this space, we note that if <math>T</math> is an ordered tree with <math>V^c</math> consisting of <math>k</math> leaves and the root, we can associate to every edge <math>e</math> in <math>T</math> a pair <math>P(e) \subset \{1,\ldots,k\}</math> called the '''node labels at <math>e</math>''':<br />
# Choose an embedding of <math>T</math> into a disk <math>D</math> in a way that respects the orders on incoming edges and which sends the critical vertices to <math>\partial D</math>.<br />
# This embedding divides <math>D^2</math> into <math>k+1</math> regions, where the 0-th region borders the root and the first leaf, for <math>1 \leq i \leq k-1</math> the <math>i</math>-th region borders the <math><br />
i</math>-th and <math>(i+1)</math>-th leaf, and the <math>k</math>-th region borders the <math>k</math>-th leaf and the root. Label these regions accordingly by <math>\{0,1,\ldots,k\}</math>.<br />
# Associate to every edge <math>e</math> in <math>T</math> the labels of the two regions which it borders.<br />
<br />
With all this setup in hand, we can finally define <math>DM(k,m;(A_j))</math>:<br />
<br />
<center><br />
<math><br />
DM(k,m;(A_j)) := \{(T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z) \;|\; \text{(1)--(6) satisfied}\}/_\sim,<br />
</math><br />
</center><br />
where (1)--(6) are defined as follows:<br />
# <math>T</math> is an ordered tree with critical vertices consisting of <math>k</math> leaves and the root.<br />
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.<br />
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order. Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.<br />
# <math>\underline\beta = (\beta_i)_{1\leq i\leq s}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-s+\sum_{1\leq i\leq s} p_i = m</math> holds.<br />
# <math>\underline z = (z_1,\ldots,z_s), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.<br />
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.<br />
<br />
The definition of the equivalence relation is defined similarly to the one in <math>DM(m)^{\text{sph}}</math>.<br />
<math>DM(k,m;(A_j))^{\text{ord}}</math> is defined similarly.<br />
<br />
== Gluing and the definition of the topology ==<br />
<br />
We will define the topology on <math>DM(k,m;(A_j))</math> (and its variants) like so:<br />
* Fix a representative <math>\hat\mu</math> of an element <math>\sigma \in DM(k,m;(A_j))</math>.<br />
* Associate to each interior nodes a gluing parameter in <math>D_\epsilon</math>; associate to each Morse-type boundary node a gluing parameter in <math>(-\epsilon,\epsilon)</math>; and associate to each Floer-type boundary node a gluing parameter in <math>[0,\epsilon)</math>. Now define a subset <math>U_\epsilon(\sigma;\hat\mu) \subset DM(k,m;(A_j))</math> by gluing using the parameters, and varying the marked points by less than <math>\epsilon</math> in Hausdorff distance.<br />
* We define the collection of all such sets <math>U_\epsilon(\sigma;\hat\mu)</math> to be a basis for the topology.<br />
<br />
To make the second bullet precise, we will need to define a '''gluing map'''<br />
<center><br />
<math><br />
[\#]: U_\epsilon(\underline 0) \times U_\epsilon(\hat\mu) \to DM(k,m;(A_j)),<br />
\quad<br />
\bigl(\underline r,<br />
</math><br />
</center><br />
<br />
== Stabilization and an example of how it's used ==<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed"><br />
old stuff<br />
<div class="mw-collapsible-content"> <br />
For <math>d\geq 2</math>, the moduli space of domains<br />
<center><math><br />
\mathcal{M}_{d+1} := \frac{\bigl\{ \Sigma_{\underline{z}} \,\big|\, \underline{z} = \{z_0,\ldots,z_d\}\in \partial D \;\text{pairwise disjoint} \bigr\} }<br />
{\Sigma_{\underline{z}} \sim \Sigma_{\underline{z}'} \;\text{iff}\; \exists \psi:\Sigma_{\underline{z}}\to \Sigma_{\underline{z}'}, \; \psi^*i=i }<br />
</math></center><br />
can be compactified to form the [[Deligne-Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>, whose boundary and corner strata can be represented by trees of polygonal domains <math>(\Sigma_v)_{v\in V}</math> with each edge <math>e=(v,w)</math> represented by two punctures <math>z^-_e\in \Sigma_v</math> and <math>z^+_e\in\Sigma_w</math>. The ''thin'' neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.<br />
<br />
The space of stable rooted metric ribbon trees, as discussed in [[https://books.google.com/books/about/Homotopy_Invariant_Algebraic_Structures.html?id=Sz5yQwAACAAJ BV]], is another topological representation of the (compactified) [[Deligne Mumford space]] <math>\overline\mathcal{M}_{d+1}</math>.<br />
</div></div></div>Natebottman