Problems on Deligne-Mumford spaces

2-, 3-dimensional associahedra

As described in Deligne-Mumford space, for any $d\geq 2$, the associahedron $\overline {\mathcal {M}}_{{d+1}}$ 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). As shown in [Auroux, Ex. 2.6], $\overline {\mathcal {M}}_{4}$ 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 $z_{2},z_{3}$. Work out which polygon/polyhedron $\overline {\mathcal {M}}_{5},\overline {\mathcal {M}}_{6}$ are equal to. (Keep in mind that when $\geq 3$ marked points collide simultaneously, there is a continuous family of ways that this collision can take place.)

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 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 [[1]]): 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.