Difference between revisions of "Problems on Deligne-Mumford spaces"
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== 2-, 3-dimensional associahedra == | == 2-, 3-dimensional associahedra == | ||
| − | As described in [[Deligne-Mumford space]], for any <math>d\geq | + | As described in [[Deligne-Mumford space]], for any <math>d\geq 2</math>, the '''associahedron''' <math>\overline\mathcal{M}_{d+1}</math> is a <math>(d-2)</math>-dimensional manifold with boundary and corners which parametrizes nodal trees of disks with <math>d+1</math> marked points, one of them distinguished (we think of the <math>d</math> undistinguished resp. 1 distinguished marked points as "input" resp. "output" marked points). |
As shown in [Auroux, Ex. 2.6], <math>\overline\mathcal{M}_4</math> is homeomorphic to a closed interval, with one endpoint corresponding to a collision of the first two inputs <math>z_1, z_2</math> and the other corresponding to a collision of <math>z_2, z_3</math>. | As shown in [Auroux, Ex. 2.6], <math>\overline\mathcal{M}_4</math> is homeomorphic to a closed interval, with one endpoint corresponding to a collision of the first two inputs <math>z_1, z_2</math> and the other corresponding to a collision of <math>z_2, z_3</math>. | ||
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. | ||
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* <math>T</math> is stable, i.e. every main vertex (=neither a leaf nor the root) has valence at least 3; | * <math>T</math> is stable, i.e. every main vertex (=neither a leaf nor the root) has valence at least 3; | ||
* <math>T</math> is a ribbon tree, i.e. the edges incident to any vertex are equipped with a cyclic ordering. | * <math>T</math> is a ribbon tree, i.e. the edges incident to any vertex are equipped with a cyclic ordering. | ||
| + | Show that the collection <math>(K_d)_{d\geq 2}</math> can be given the structure of an '''operad''', which is to say that for every <math>d, e \geq 2</math> and <math>1 \leq i \leq d</math> there is a composition operation <math>\circ_i\colon K_d \times K_e \to K_{d+e-1}</math> which splices <math>T_e \in K_e</math> onto <math>T_d \in K_d</math> by identifying the outgoing edge of <math>T_e</math> with the <math>i</math>-th incoming edge of <math>T_d</math>. | ||
Revision as of 14:59, 26 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.)
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.
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 
.
