Difference between revisions of "Problems on Deligne-Mumford spaces"

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Work out which polygon/polyhedron <math>\overline\mathcal{M}_5, \overline\mathcal{M}_6</math> are equal to.
 
Work out which polygon/polyhedron <math>\overline\mathcal{M}_5, \overline\mathcal{M}_6</math> are equal to.
 
(Keep in mind that when <math>\geq 3</math> marked points collide simultaneously, there is a continuous family of ways that this collision can take place.)
 
(Keep in mind that when <math>\geq 3</math> marked points collide simultaneously, there is a continuous family of ways that this collision can take place.)
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 +
== 2-, 3-dimensional examples of "the original" Deligne-Mumford space ==
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Define <math>\mathcal{M}_{0,d+1}(\mathbb{C})</math> to be the moduli space
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<center><math>
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\mathcal{M}_{0,d+1}(\mathbb{C}) := \bigl\{ \underline{z} = \{z_0,\ldots,z_d\}\in \mathbb{CP}^1 \;\text{pairwise disjoint} \bigr\}/_\sim,
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</math></center>
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where two configurations are identified if one can be taken to the other by a Moebius transformation.
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We can compactify to form <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C})</math> by including trees of spheres, any neighboring pair of which is attached to one another at a pair of points.
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(A detailed construction of <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C})</math> can be found in [big McDuff-Salamon, App. D].)
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Make the identifications (don't worry too much about rigor) <math>\overline\mathcal{M}_{0,3}(\mathbb{C}) \cong \mathrm{pt}</math>, <math>\overline\mathcal{M}_{0,4}(\mathbb{C}) \cong \mathbb{CP}^1</math>, and <math>\overline\mathcal{M}_{0,5}(\mathbb{C}) \cong (\mathbb{CP}^1\times\mathbb{CP}^1) \# 3\overline{\mathbb{CP}}^2</math>.
  
 
== poset underlying associahedra ==
 
== poset underlying associahedra ==

Revision as of 05:13, 27 May 2017


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 trees of spheres, any neighboring pair of which is attached to one another 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.

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