Problems on Deligne-Mumford spaces

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These problems deal with Deligne-Mumford spaces, by which we mean the moduli spaces of domains relevant for our construction of the Fukaya category. The first two problems form the warm-up portion: you should make sure you can do these before Tuesday morning. The remaining two problems form the further fun section: useful for deeper understanding, but not essential for following the thread of the lectures.

2-, 3-dimensional associahedra

Using the notation of Deligne-Mumford space, for any d\geq 2, the associahedron \overline {\mathcal  {M}}_{{d+1}}:=DM(d+1;0)^{0} is a (d-2)-dimensional manifold with boundary and corners which parametrizes nodal trees of disks with d+1 marked points, one of them distinguished (we think of the d undistinguished resp. 1 distinguished marked points as "input" resp. "output" marked points). As shown in [Auroux, Ex. 2.6 [[1]]], \overline {\mathcal  {M}}_{4} is homeomorphic to a closed interval, with one endpoint corresponding to a collision of the first two inputs z_{1},z_{2} and the other corresponding to a collision of z_{2},z_{3}. Moving up a dimension, \overline {\mathcal  {M}}_{5} is a pentagon; it's the central pentagon in the depiction of DM(5;0) in Deligne-Mumford space. Work out which polygon/polyhedron \overline {\mathcal  {M}}_{5},\overline {\mathcal  {M}}_{6} are equal to. (A good way to get started on this problem is to list the codimension-1 strata.)

The associahedra are similar to the Deligne-Mumford spaces we will use during the summer school. The difference is that we will augment the associahedra by labeling the interior edges of the underlying tree by elements of [0,1], and allow (unordered) interior marked points.

2-, 3-dimensional examples of "the original" Deligne-Mumford space

Define {\mathcal  {M}}_{{0,d+1}}({\mathbb  {C}}) to be the moduli space

{\mathcal  {M}}_{{0,d+1}}({\mathbb  {C}}):={\bigl \{}\underline {z}=\{z_{0},\ldots ,z_{d}\}\in {\mathbb  {CP}}^{1}\;{\text{pairwise disjoint}}{\bigr \}}/_{\sim },

where two configurations are identified if one can be taken to the other by a Moebius transformation. We can compactify to form \overline {\mathcal  {M}}_{{0,d+1}}({\mathbb  {C}}) 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. (A detailed construction of \overline {\mathcal  {M}}_{{0,d+1}}({\mathbb  {C}}) can be found in [big McDuff-Salamon, App. D].) Make the identifications (don't worry too much about rigor) \overline {\mathcal  {M}}_{{0,3}}({\mathbb  {C}})\cong {\mathrm  {pt}}, \overline {\mathcal  {M}}_{{0,4}}({\mathbb  {C}})\cong {\mathbb  {CP}}^{1}, and \overline {\mathcal  {M}}_{{0,5}}({\mathbb  {C}})\cong ({\mathbb  {CP}}^{1}\times {\mathbb  {CP}}^{1})\#3\overline {{\mathbb  {CP}}}^{2}.

poset underlying associahedra

The associahedron \overline {\mathcal  {M}}_{{d+1}} can be given the structure of a stratified space, where the underlying poset is called K_{d} and consists of stable rooted ribbon trees with d leaves. Similarly to the setup in Moduli spaces of pseudoholomorphic polygons, a stable rooted ribbon tree is a tree T satisfying these properties:

  • T has d leaves and 1 root (in our terminology, the root has valence 1 but is not counted as a leaf);
  • T is stable, i.e. every main vertex (=neither a leaf nor the root) has valence at least 3;
  • T is a ribbon tree, i.e. the edges incident to any vertex are equipped with a cyclic ordering.

To define the partial order, we declare T'\leq T if we can contract some of the interior edges in T' to get T; we declare that T' is in the closure of T if T'\leq T. Write the closure of the stratum corresponding to T as a product of lower-dimensional K_{d}'s. Which tree corresponds to the top stratum of \overline {\mathcal  {M}}_{{d+1}}? To the codimension-1 strata of \overline {\mathcal  {M}}_{{d+1}}?

...and, to the operadically initiated (or willing to dig around a little at [[2]]): show that the collection (K_{d})_{{d\geq 2}} can be given the structure of an operad (which is to say that for every d,e\geq 2 and 1\leq i\leq d there is a composition operation \circ _{i}\colon K_{d}\times K_{e}\to K_{{d+e-1}} which splices T_{e}\in K_{e} onto T_{d}\in K_{d} by identifying the outgoing edge of T_{e} with the i-th incoming edge of T_{d}, and that these operations satisfy some coherence conditions). Next, show that algebras / categories over the operad (C_{*}(K_{d}))_{{d\geq 2}} of cellular chains on K_{d} are the same thing as A_{\infty } algebra / categories.