Problems on Deligne-Mumford spaces
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 , 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 [[1]]], 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 . Moving up a dimension, is a pentagon; it's the central pentagon in the depiction of in Deligne-Mumford space. Work out which polygon/polyhedron 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 , and allow (unordered) interior marked points.
2-, 3-dimensional examples of "the original" Deligne-Mumford space
Define to be the moduli space
where two configurations are identified if one can be taken to the other by a Moebius transformation. We can compactify to form 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 can be found in [big McDuff-Salamon, App. D].) Make the identifications (don't worry too much about rigor) , , and .
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.
To define the partial order, we declare if we can contract some of the interior edges in to get ; we declare that is in the closure of if . 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 [[2]]): 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.