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.) | ||
+ | |||
+ | == 2-, 3-dimensional examples of "the original" Deligne-Mumford space == | ||
+ | |||
+ | Define <math>\mathcal{M}_{0,d+1}(\mathbb{C})</math> to be the moduli space | ||
+ | <center><math> | ||
+ | \mathcal{M}_{0,d+1}(\mathbb{C}) := \bigl\{ \underline{z} = \{z_0,\ldots,z_d\}\in \mathbb{CP}^1 \;\text{pairwise disjoint} \bigr\}/_\sim, | ||
+ | </math></center> | ||
+ | where two configurations are identified if one can be taken to the other by a Moebius transformation. | ||
+ | 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. | ||
+ | (A detailed construction of <math>\overline\mathcal{M}_{0,d+1}(\mathbb{C})</math> can be found in [big McDuff-Salamon, App. D].) | ||
+ | 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 06:13, 27 May 2017
2-, 3-dimensional associahedra
As described in 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).
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
.
Work out which polygon/polyhedron
are equal to.
(Keep in mind that when
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 to be the moduli space

where two configurations are identified if one can be taken to the other by a Moebius transformation.
We can compactify to form by including trees of spheres, any neighboring pair of which is attached to one another 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
.
poset underlying 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 [[1]]): 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.