Difference between revisions of "Problems on Deligne-Mumford spaces"
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...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). | ...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). | ||
| − | Next, show that algebras / categories | + | 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. |
Revision as of 16:09, 26 May 2017
2-, 3-dimensional associahedra
As described in Deligne-Mumford space, for any 
, the associahedron 
is a 
-dimensional manifold with boundary and corners which parametrizes nodal trees of disks with 
marked points, one of them distinguished (we think of the 
undistinguished resp. 1 distinguished marked points as "input" resp. "output" marked points).
As shown in [Auroux, Ex. 2.6], 
is homeomorphic to a closed interval, with one endpoint corresponding to a collision of the first two inputs 
and the other corresponding to a collision of 
.
Work out which polygon/polyhedron 
are equal to.
(Keep in mind that when 
marked points collide simultaneously, there is a continuous family of ways that this collision can take place.)
poset underlying associahedra
The associahedron can be given the structure of a stratified space, where the underlying poset is called 
and consists of stable rooted ribbon trees with leaves.
Similarly to the setup in Moduli spaces of pseudoholomorphic polygons, a stable rooted ribbon tree is a tree 
satisfying these properties:
- has leaves and 1 root (in our terminology, the root has valence 1 but is not counted as a leaf);
- is stable, i.e. every main vertex (=neither a leaf nor the root) has valence at least 3;
- is a ribbon tree, i.e. the edges incident to any vertex are equipped with a cyclic ordering.
Write the closure of the stratum corresponding to as a product of lower-dimensional 's.
Which tree corresponds to the top stratum of ?
To the codimension-1 strata of ?
...and, to the operadically initiated (or willing to dig around a little at [[1]]): show that the collection 
can be given the structure of an operad (which is to say that for every 
and 
there is a composition operation 
which splices 
onto 
by identifying the outgoing edge of 
with the 
-th incoming edge of 
, and that these operations satisfy some coherence conditions).
Next, show that algebras / categories over the operad 
of cellular chains on are the same thing as 
algebra / categories.
