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 problem forms the warm-up portion: you should make sure you understand how to do this one 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.
Down and dirty with low-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). 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.
(a) 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 . Moving up a dimension, is a pentagon; it can be identified with the central pentagon in the depiction of in Deligne-Mumford space. Which polyhedron is equal to? (A good way to get started on this problem is to list the codimension-1 strata.)
(b) Using the manifold-with-corners structure of the associahedra constructed in Deligne-Mumford space, observe that the -dimensional associahedron can be covered by three charts:
- boundary charts centered respectively at the two points in (the domains of these charts are of the form , and a choice of a point in this interval tells us how much to smooth the node);
- an interior chart (which we produce by fixing the positions of three of the marked points, and varying the position of the fourth).
Explicitly work out these charts, and the transition maps amongst them.
The poset indexing the strata of the 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 []): 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.
More low-dimensional examples of Deligne-Mumford spaces
(a) Explicitly work out the 3-dimensional Deligne-Mumford space .
(b) For define . That is, if we define
where two configurations are identified if one can be taken to the other by a Moebius transformation, then 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. (A detailed construction of can be found in [big McDuff-Salamon, App. D].) Make the identifications (don't worry too much about rigor) , , and .