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 $d\geq 2$, the associahedron $\overline {\mathcal {M}}_{{d+1}}:=DM(d,0,(\{0\},\ldots ,\{d\}))$ 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 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 [[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 can be identified with the central pentagon in the depiction of $DM(4,0;(\{0,1,2,3,4\}))$ 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 $\overline {\mathcal {M}}_{3}$ can be covered by two charts, centered at the two points in $\overline {\mathcal {M}}_{3}\setminus {\mathcal {M}}_{3}$; 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.

The poset indexing the strata of the 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.

More low-dimensional examples of Deligne-Mumford spaces

(a) Explicitly work out the 3-dimensional Deligne-Mumford space $DM(4;1)^{0}$.

(b) For $d\geq 2$ define $\overline {\mathcal {M}}_{{0,d+1}}({\mathbb {C}}):=DM(d)^{{{\text{sph,ord}}}}$. That is, if we define

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

where two configurations are identified if one can be taken to the other by a Moebius transformation, then $\overline {\mathcal {M}}_{{0,d+1}}({\mathbb {C}})$ 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 $\overline {\mathcal {M}}_{{0,d+1}}({\mathbb {C}})$ can be found in [big McDuff-Salamon, App. D].) Make the identifications (don't worry too much about rigor) $\overline {\mathcal {M}}_{{0,4}}({\mathbb {C}})\cong {\mathrm {pt}}$, $\overline {\mathcal {M}}_{{0,5}}({\mathbb {C}})\cong {\mathbb {CP}}^{1}$, and $\overline {\mathcal {M}}_{{0,6}}({\mathbb {C}})\cong ({\mathbb {CP}}^{1}\times {\mathbb {CP}}^{1})\#3\overline {{\mathbb {CP}}}^{2}$.