# Tagged Questions

**8**

votes

**0**answers

251 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 ...

**5**

votes

**1**answer

309 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 ...

**16**

votes

**1**answer

543 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 ...

**9**

votes

**4**answers

877 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 ...

**12**

votes

**3**answers

607 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 ...

**18**

votes

**4**answers

1k 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 ...

**2**

votes

**1**answer

771 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} ...

**4**

votes

**1**answer

591 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 ...

**5**

votes

**1**answer

510 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 ...

**0**

votes

**2**answers

221 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 ...

**15**

votes

**0**answers

346 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). ...

**10**

votes

**1**answer

565 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 ...

**15**

votes

**4**answers

733 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 ...

**6**

votes

**1**answer

511 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, $X^{[n]}, \Sigma^{[n]}$ ...

**11**

votes

**4**answers

1k 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 ...

**25**

votes

**3**answers

2k 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 ...

**8**

votes

**5**answers

2k 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?