All Questions
Tagged with at.algebraic-topology moduli-spaces
31 questions
30
votes
3
answers
3k
views
Mumford conjecture: Heuristic reasons? Generalizations? ... Algebraic geometry approaches?
The Mumford conjecture states that for each integer $n$, we have: the map $\mathbb{Q}[x_1,x_2,\dots] \to H^\ast(M_g ; \mathbb{Q})$ sending $x_i$ to the kappa class $\kappa_i$, is an isomorphism in ...
- 21k
22
votes
4
answers
3k
views
Betti numbers of moduli spaces of smooth Riemann surfaces
Where can I find a list of the known Betti numbers of the moduli spaces $\mathcal{M}_{g,n}$ of genus $g$ Riemann surfaces with $n$ marked points? I need it to cross check results by an implemented ...
- 6,777
19
votes
4
answers
2k
views
Details for the action of the braid group B_3 on modular forms
I'm reading Terry Gannon's Moonshine Beyond the Monster, and in section 2.4.3 he hints at (but does not explicitly describe) a way to extend the action of $SL_2(\mathbb{Z})$ on modular forms to an ...
- 118k
19
votes
0
answers
504
views
Other examples of computations using transfer of structure from the chains to the homology?
There is a `long' history of transfer (up to homotopy!) of algebraic structure from a dg _ algebra A to its homology H(A) (e.g. Kadeishvili for the associative case and Heubschmann for the Lie case). ...
- 3,880
18
votes
1
answer
797
views
Do mapping classes have gonality?
(This question was discussed by people at the PCMI workshop on moduli spaces, without any clear resolution, so I thought I'd throw it open to MO.)
The hyperelliptic mapping class group is (by ...
- 19.2k
17
votes
4
answers
2k
views
What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve?
Let $X$ be a smooth surface of genus $g$ and $S^nX$ its n-symmetrical product (that is, the quotient of $X \times ... \times X$ by the symmetric group $S_n$). There is a well known, cool formula ...
- 445
17
votes
0
answers
553
views
Lie algebras vs. graph complexes
A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain ...
- 2,035
16
votes
1
answer
360
views
Moduli space of boundary maps with prescribed chain and homology groups?
Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C_n$ and $H_n$ of $R$-modules, which are tame in every ...
- 15.5k
15
votes
3
answers
1k
views
Are there graph models for other moduli spaces?
Recall that a ribbon graph is a graph with a cyclic ordering at each vertex and such that each vertex has valence greater than or equal to 3. This cyclic ordering exactly gives one the information to ...
- 8,167
13
votes
4
answers
2k
views
Why does (Ribbon) Graph (co)Homology Compute (co)Homology of MCG?
The title says it all. I am looking for an explanation or reference for why the homology of the ribbon graph complex computes the cohomology of the mapping class groups of surfaces.
I've seen ...
- 2,283
12
votes
2
answers
990
views
A-infinity structure on the ribbon graph complex and more general graph complexes
Moduli spaces of curves (with nonempty boundary or at least one marked point) admit cell decompositions in which the cells are labelled by ribbon graphs. In fact, the moduli space of normalised ...
- 6,229
11
votes
5
answers
3k
views
Ribbon graph decomposition of the moduli space of curves
What is a ribbon graph? What is the ribbon graph decomposition of the moduli space of curves? What are some good references for this material?
- 21k
8
votes
1
answer
529
views
Topology on the space of constructible sheaves
Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with ...
- 21.8k
7
votes
0
answers
312
views
Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?
We are taught that, in general:
A type of objects that has nontrivial automorphisms cannot have a fine moduli space.
The proof generally goes along the lines of:
Take an object $X$ with a non-...
- 907
6
votes
1
answer
675
views
Some questions on the intersection theory on a Hilbert scheme of points of a surface.
If $\Sigma$ is a smooth complex curve in a smooth projective surface $X$, then we can consider the homology class represented by $\Sigma^{[n]} \subset X^{[n]}$. $\ \ $ Where, $X^{[n]}, \Sigma^{[n]}$ ...
- 445
5
votes
1
answer
436
views
Is $M_{1,n}$ affine?
A famous conjecture of Looijenga states that the moduli space of curves $M_{g,n}$ is the union of $g- \delta_{0,n}+ \delta_{0,g}$ open affine subsets, where $g,n$ are non-negative integers satisfying $...
5
votes
1
answer
976
views
How can we show the spaces $M_{g}(n)$ and $M_{g, n}$ are homotopy equivalent?
How can we prove that the moduli space,$M_{g}(n)$, of genus $g$ Riemann surface with $n$ boundary components is homotopy equivalent to $M_{g,n}$, that is ,the moduli space of genus $g$ Riemann surface ...
- 1,499
5
votes
1
answer
344
views
Is there a relationship between the moduli space of spatial polygons and the moduli space of labeled points?
It is well known that the set of all polygons with consecutive side lengths $l_1, \dots, l_n$ in $\mathbb{R}^3$, considered up to rigid motions, is a compact complex manifold. Of course, I am assuming,...
- 1,017
5
votes
0
answers
175
views
Is $\overline{\mathcal{M}}_{g,n}$ a Koszul space?
In https://arxiv.org/abs/1902.06318 Dotsenko proved that $\overline{\mathcal{M}}_{0,n+1}$ is a Koszul space, i.e. it is both formal and coformal. Equivalently, a space is Koszul if it is formal and ...
- 641
5
votes
0
answers
416
views
Topology of the space of foliations on a 3-manifold
Denote by $\mathcal{P} (M)$ the space of smooth plane fields(oriented and transversely oriented) on a given closed and orientable 3-manifold $M$ with the $C^{\infty}$ topology, and by $\mathcal{F}(M)$ ...
- 858
4
votes
1
answer
195
views
Stable cohomology of mapping class group with coefficients in $H^{\otimes n}$
Let $\text{Mod}_g$ be the mapping class group of a closed oriented genus-$g$ surface $\Sigma_g$ and let $H = H_1(\Sigma_g;\mathbb{Q})$. Fix some $r \geq 0$. It is known that the cohomology group $H^...
- 43
4
votes
1
answer
2k
views
On Thurston's triangulations of sphere
I have two questions from Thurston's paper [1].
In the paper [1], Thurston talks about classifying certain classes of triangulations of the sphere. Here a triangulation of a sphere a Topological ...
- 239
4
votes
1
answer
805
views
Homology dimension of the mapping class group of a surface with boundary
There is a result on the dimension bound for ${M_{g,n}}/S_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is
$H_{i}({M_{g,n}}/S_n)=0$, for $i\ge 6g-7+2n$ except $(...
- 1,499
4
votes
0
answers
98
views
Nakamura graphs and moduli space cellular decomposition
I have recently started studying the cell decomposition of moduli spaces. Among the papers I read, I studied this paper, but there is something I do not understand and I can't find the answer on my ...
- 641
3
votes
0
answers
126
views
degenerate points in the moduli space of flat principal $G$-bundle with respect to a linear representation on a complex
Let $(A^\bullet,\partial)$ be a complexe of $\mathbb{C}$-vector spaces. We suppose that this complex is of finite length, and all $A^\bullet$ are finite dimensional. Let $H^\bullet$ be the cohomology ...
- 101
2
votes
1
answer
1k
views
Question related to the moduli space of Riemann surfaces and a fibration
If $M_{g}$ is the moduli space of Riemann surface of genus $g$, $M^1_{g}$ is the moduli space of Riemann surface of genus $g$ with one boundary, how can we show that the natural map:
$M^1_{g} \...
- 1,499
2
votes
1
answer
1k
views
Moduli space of flat connections over a torus
Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, ...
- 2,816
1
vote
1
answer
232
views
Pullback of boundary divisors under forgetful maps
Let $\overline{\mathbf{M}}_{0,n}$ be the moduli space of stable $n-$pointed smooth rational curve of genus zero and $\overline{\mathbf{U}}_{0,n}$ the universal family described by $\pi_n:\overline{\...
- 31
1
vote
0
answers
120
views
Local description of the universal family $\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}$
I would like to get an understanding of the notion of geometric fibers of the universal family:
$$\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}.$$
In fact Knudsen show ...
- 31
0
votes
2
answers
295
views
Does the Deligne-Mumford space module $S_{n}$ action have a fundamental chain?
Does the Deligne-Mumford space (without ordering for marked points) $\bar M_{g,n}/S_{n}$ has fundamental chain in signular simplicial chains? (because I read Costello's paper GW potential to TCFT, as ...
- 285
0
votes
1
answer
731
views
Generalizing the Madsen-Weiss Theorem via the scanning map $\mathscr{C}(M,\mathbb{R}^{\infty})\to\Omega^{\infty}AG^+_{\infty,d}$
The Madsen-Weiss Theorem, as described by Hatcher, states that there is an isomorphism $H_*( \mathscr{C}_{\infty})\cong H_*(\Omega_0^{\infty}AG^+_{\infty,2})$ where $\Omega_0^{\infty}AG^+_{\infty,2}$ ...
- 239