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.

Down and dirty with low-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). 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.

(a) 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. Which polyhedron is $\overline {\mathcal {M}}_{6}$ 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 $\overline {\mathcal {M}}_{4}$ can be covered by two charts, centered at the two points in $\overline {\mathcal {M}}_{4}\setminus {\mathcal {M}}_{4}$; the domains of these charts are of the form $[0,a)$, and a choice of a number in this interval tells us how much to smooth the node. Explicitly work out the transition map between these charts.

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.