Betti numbers of moduli spaces of smooth Riemann surfaces

Question

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 algorithm which should be producing them using Kontevich's graph complex.

I am interested in the "open" moduli space consisting of smooth connected surfaces, not in its Deligne-Mumford compactification $\overline{\mathcal{M}}_{g,n}$. Also, I'm interested in the single Betti numbers and not in the Euler characteristic, which I know from e.g. Harar-Zagier and Bini-Gaiffi-Polito, and which I used to have a first check of the results of the algorithm.

Thanks.

Edit: Riccardo Murri's paper with the algorithm and its implementation has now appeared on arXiv: https://arxiv.org/abs/1202.1820

Answer 1

Calculations of integral homology of $\mathcal{M}_{g, n}$ occur in Abhau, Bodigheimer, Ehrenfried (p. 3) or Godin (p. 4). Of course, in the stable range (degrees $3* \leq 2g$) the rational cohomology is entirely known (by Mumford's conjecture, now theorem), and is a polynomial algebra on a single generator in each even degree. Thus the Betti numbers in this range are given by partition functions.

I should mention that Godin's results were obtained using the complex of fat graphs, which is probably equal to Kontsevich's graph complex for the associative operad.

Answer 2

The references given by Oscar Randal-Willams only cover $g\leq 2$. For $\mathcal{M}$$_3$ and $\mathcal{M}_{3,1}$, you can read Looijenga, "Cohomology of $\mathcal{M}$$_3$ and $\mathcal{M}^1_3$".

For $\mathcal{M}$$_{2,4}$, $\mathcal{M}_{3,2}$ and $\mathcal{M}_4$ the Betti numbers can be found in Bergström–Tommasi, "The rational cohomology of $\overline{\mathcal{M}}_4$" (Math. Annalen version, arXiv version), sometimes summarizing their individual papers. For example, for $\mathcal{M}_4$, Tommasi ("Rational cohomology of the moduli space of genus 4 curves") proved that $b_1=0$, $b_2=1$, $b_3=0$, $b_4=1$, $b_5=1$, and the rest are 0.

You haven't specified whether your marked points are individually labeled, or if they are allowed to be permuted. Fortunately Bergström–Tommasi compute the $S_n$--equivariant cohomology, so you can extract from that paper whichever answer you want.

One remark: The formula for the stable Betti numbers mentioned by Oscar Randal-Williams is for closed surfaces; the formula for marked points will be different. I think the stable cohomology for surfaces with marked points can probably be deduced either from Mumford's conjecture, or from one of the many proofs we now have. But I do not know where this has been written down.

Answer 3

Complementing other answers in this thread:

First, when $n>0$, there is the Penner decomposition (see e.g. Harer, The cohomology of the moduli space of curves, LNM 1337 or Penner, Comm Math Phys 113, 299-339). This gives in principle a finite-dimensional complex that computes the cohomology of the coarse moduli spaces. In practice however the number of cells quickly gets quite large and I'm not sure whether or not someone has written a program that implements this.

When $g=0$, the cohomology was computed by S. Keel (Transactions AMS, 330, 2, 545-574) for any number of points. Keel also computes the action of the symmetric groups, the cup product and the rational mixed Hodge structure, which turns out to be a direct sum of Tate ones.

[upd: Keel is primarily interested in the cohomology of the Deligne-Mumford compactification and it is not clear to me how to deduce the cohomology of the open part from his results. So here is an ad hoc way to describe the cohomology of $M_{0,n}$. Let $H=H^*(\mathbf{P}^1(\mathbf{C}),\mathbf{Q})$. Following B. Totaro (Configuration spaces of algebraic varieties, Topology 35 (1996), no. 4, 1057--1067) form the cdg-algebra $$F=H^{\otimes n}[a_{i,j}]/rels$$ where $a_{i,j},i,j=1,\ldots, n, i\neq j$ are variables in degree 1 (so they anti-commute with everything) and the relations $rels$ are

1. $a_{i,j}=a_{j,i}$;

2. the cyclic permutations of $a_{i,j}a_{j,k}$ add up to 0 where $i,j,k$ are pairwise distinct;

3. $a_{i,j}(h_i-h_j)=0$ where $h_i=1\otimes\ldots\otimes h\otimes\ldots\otimes 1$ ($h\in H$ in the $i$-th place), and similarly for $h_j$.

The differential annihilates $H^{\otimes n}$ and takes $a_{i,j}$ to the pullback of the class of the diagonal under the projection $p_{i,j}:\mathbf{P}^1(\mathbf{C})^{\times n}\to \mathbf{P}^1(\mathbf{C})^{\times 2}$ to the $i$-th and $j$-th factors.

The cohomology of $F$ is the cohomology of the space $F(\mathbf{P}^1(\mathbf{C}),n)$ of ordered $n$-tuples of distinct points in $\mathbf{P}^1(\mathbf{C})$. The moduli space is the quotient of $F(\mathbf{P}^1(\mathbf{C}),n)$ by the action of $PGL_2(\mathbf{C})$. Now for this action the Leray-Hirsch principle holds: there is a degree 3 class whose restriction to each orbit generates the $H^3$ of the orbit. To see this note that $F(\mathbf{P}^1(\mathbf{C}),3)\cong PGL_2(\mathbf{C})$. If we take a generator of $H^3( F(\mathbf{P}^1(\mathbf{C}),3),\mathbf{Q})$ and take the sum of its pullbacks under all possible projections $F(\mathbf{P}^1(\mathbf{C}),n)\to F(\mathbf{P}^1(\mathbf{C}),3)$, this should do the trick. So $$H^{\ast}(F(\mathbf{P}^1(\mathbf{C}),n))\cong H^{\ast}(PGL_2(\mathbf{C}))\otimes H^{\ast}(M_{0,n})$$ with $\mathbf{Q}$-coefficients.

So the recipe to compute say the complex the cohomology of $M_{0,n}$ as an $S_n$-module is as follows: form a polynomial $f(t)=\sum c_i t^i$ where $c_i=H^i(F)$ viewed as an element of the representation ring $R(S_n)$ of $S_n$. This polynomial is the product of $1+t^3$ and some other polynomial $g$, which will be the $S_n$-equivariant Poincar'e polynomial of $M_{0,n}$.

Note also that if one is interested only in the Poincar'e polynomial and not in the action of $S_n$, then the answer is simply $(1+2t)(1+3t)\cdots (1+(n-2)t)$.

I'm not sure though what the reference for this is or whether there is a better description of $H^*(M_{0,n},\mathbf{Q})$ or whether the answer has been tabulated for small $n$. I'd be interested to know the answer to either of these questions.]

Let me also mention two results on the Euler characteristics that extend Harer-Zagier. Bini and Harer give an explicit formula for the Euler characteristic of the Deligne-Mumford compactified moduli spaces in https://arxiv.org/abs/math/0506083; E. Gorsky https://arxiv.org/abs/0906.0841 computes the $S_n$-equivariant Euler characteristic of $M_{g,n}$ for an arbitrary $n$. By Getzler-Kapranov this also gives the equivariant Euler characteristic of the compactified moduli spaces.

Answer 4

Complementing the earlier answers:

In genus zero, a very useful reference is Ezra Getzler, "Operads and moduli spaces of genus 0 Riemann surfaces". He determines generating series for the $\Sigma_n$-equivariant Poincaré polynomials of $\overline M_{0,n}$ and $M_{0,n}$, and proves the purity of the cohomology of $M_{0,n}$: the cohomology group $H^i(M_{0,n})$ carries a pure Hodge structure of weight $2i$ (of Tate type).

Also two small remarks on algori's answer. (i) If I am not mistaken, the quotient map $F(\mathbf P^1,n) \to M_{0,n}$ is in fact a trivial $\mathrm{PGL}_2$-bundle, since it has a section given by putting the first three points at $0$, $1$ and $\infty$. So things are even simpler. (ii) If you don't care about the $\Sigma_n$-equivariance, then you can use exactly the same tool (the Cohen-Taylor-Totaro spectral sequence) but consider instead the configuration space $$ M_{0,n} \cong F(\mathbf P^1 \setminus \{0,1,\infty\},n-3). $$ Since $\mathbf P^1$ minus three points has $H^0$ of weight $0$ and $H^1$ of weight $2$, and the differentials are compatible with weights, it follows that the spectral sequence degenerates immediately so one can directly read off the Betti numbers and the purity.

In genus one, there is a recent preprint of Gorinov that determines the cohomology of $M_{1,n}$ with its Hodge structure. I believe that what he does is the following:

Consider the configuration space of $n$ points on an elliptic curve, $F(E,n)$, quotiented by the action of $E$ by translation. The cohomology of this space can be computed via the Cohen-Taylor-Totaro spectral sequence. If you are careful you can even determine how the cohomology of $F(E,n)/E$ is built out of (symmetric powers of) $H^1(E)$ and some Tate twists.

Now consider the forgetful map $\pi \colon M_{1,n}\to M_{1,1}$. The results of the previous paragraph tell you how the local systems $R^i \pi_\ast \mathbf Q$ are built out of symmetric powers of the "standard local system", which is $R^1\pi_\ast \mathbf Q$ for $n=1$. Finally, the cohomology of these local systems is given by Eichler-Shimura theory; the cohomology groups are canonically given by spaces of modular forms for $\mathrm{SL}(2,\mathbf Z)$. As far as I can tell, the Eichler-Shimura theory is the only part which is specific to genus one: everything else would work without changes in higher genus as well.

Maybe I should mention at this point that there is plenty to read about how to compute the cohomology of these local systems on $M_g$ when $g \geq 2$. For $g=2$ and $g=3$ you can read a sequence of papers of Faber, van der Geer and Bergström, who work on computing/conjecturing Euler characteristics of these local systems in the Grothendieck group of $\ell$-adic Galois representations by means of point counts. The Euler characteristic doesn't give you the cohomology, but it gives you partial information. In particular because you know that these local systems are pulled back from $A_g$, and on $A_g$, most cohomology groups of these local systems vanish. (There is a wealth of information in Faltings-Chai, chapter 6.) Then you can apply the Gysin exact sequence for the image of $M_g$ in $A_g$ under the Torelli map.

Let me finally make a few remarks on genus two. If you only want the Betti numbers (i.e. you don't care about torsion), then there are easier ways than the papers linked by Oscar Randal-Williams. When $n=0$, there is an isomorphism $M_2 \cong M_{0,6}/\Sigma_6$ on coarse moduli spaces: every curve of genus two is hyperelliptic, and a hyperelliptic curve is uniquely determined by the $2g+2$ branch points on $\mathbf P^1$ under the hyperelliptic map. So rationally, you need only to take the $\Sigma_6$-invariants in Getzler's description of the cohomology of $M_{0,n}$, and you find that $M_2$ has the rational cohomology of a point.

When $n=1$, you apply the Leray spectral sequence for $\pi \colon M_{2,1} \to M_2$. The local systems $R^0\pi_\ast\mathbf Q$ and $R^2\pi_\ast\mathbf Q$ are trivial, and $R^1\pi_\ast\mathbf Q$ has no cohomology since every point of $M_2$ has the hyperelliptic involution in its automorphism group, which acts as multiplication by $-1$ on the fibers of the latter local system (this also uses that you take rational coefficients). You conclude that $H^\ast(M_{2,1}) \cong H^\ast(\mathbf P^1)$.

Note that this generalizes to hyperelliptic curves of any genus: you always have that $H_g$ has the rational cohomology of a point, and $H_{g,1}$ the cohomology of $\mathbf P^1$.

For $n=2$ you need to compute the cohomology of the local systems $V_{2}$ and $V_{1,1}$ on $M_2$. Getzler manages to determine all but two Betti numbers without this information in "Topological recursion relations in genus two" and expresses the final Betti numbers in terms of their cohomology. The cohomology of these local systems can be found in the more recent paper by Hulek and Tommasi, "Cohomology of the second Voronoi compactification of $A_4$", Appendix A. (This appendix also contains some things useful for $M_3$.) Together these papers determine the Betti numbers of $M_{2,2}$. I would guess that you can compute also the Betti numbers of $M_{2,3}$ from this information, since the only "new" local systems that appear for $n=3$ have odd weight so their cohomology vanishes, but I have not sat down to compute the Leray spectral sequence here.

The Betti numbers for $M_{2,4}$ I suspect are unknown. Tom Church writes above that they can be found in Bergström-Tommasi, but I think this is a misreading of their paper. As Tom writes, much of the paper of Bergström and Tommasi summarizes their previous work, done separately and by different methods. Tommasi's work uses the Vassiliev-Gorinov method of computing the cohomology of complements of discriminants. This gives you the Poincaré-Serre polynomial, and in particular the Betti numbers. Bergström uses point counts over finite fields, which gives you the Euler characteristic in the category of $\ell$-adic Galois representations (or what they call the Hodge Euler characteristic), but NOT in general the Betti numbers. The results on $M_{2,n}$ are due to Bergström in "Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves".