Is there Harer stability for moduli of curves with level structure?

Question

The famous Harer stability theorem asserts that the homology group $H_d(\mathcal{M}_{g,n},\mathbf{Z})$ is independent of g and n in the range $0 \leq 2d < g-1$. This is proven by analyzing the maps of mapping class groups $\Gamma_{g,n}\to \Gamma_{g+1,n}$ given by gluing a torus with a disk removed to a boundary circle (when $n \geq 1$), and $\Gamma_{g,n} \to \Gamma_{g,n-1}$ by gluing in a disk, and showing that these maps induce an isomorphism on homology in low dimensions (regardless of the choices involved in writing down such maps).

By considering curves with level structures, one obtains finite covers of $\mathcal{M}_{g,n}$, or equivalently, finite index subgroups of the mapping class group. So let's consider a finite group G, and denote by $\mathcal{M}_{g,n}[G]$ the moduli space parametrizing n-pointed smooth curves of genus g equipped with an étale G-torsor. Is the corresponding statement for $H_d(\mathcal{M}_{g,n}[G],\mathbf{Z})$ known? It is not hard to write down analogues in this context of the maps of mapping class groups above.

Remark: The corresponding statement for moduli of spin curves is known (and is also a theorem of Harer), so one might hope for a statement like this because of the similarities between the spaces of r-spin curves and the spaces of curves with $G=\mathbf{Z}/r\mathbf{Z}$ level structure.

Answer 1

This is a hard open problem. Essentially nothing is known except for linear congruence subgroups. Denoting by $Mod_{g,n}(L)$ the level $L$ linear congruence subgroup, the desired result is only known for $H_1(Mod_{g,n}(L);\mathbb{Q})$ (which is due to Hain) and for $H_2(Mod_{g,n}(L);\mathbb{Q})$ (which is due to me). See my paper "The second rational homology groups of the moduli space of curves with level structures" (available on my webpage).

I should also remark that the corresponding result is false if you replace $\mathbb{Q}$ with $\mathbb{Z}$, even for $H_1$. See Theorem F of my paper "The Picard group of the moduli space of curves with level structures".

One observation that is worth making (and I make it in my paper "The second rational homology...") is that if $H_k(Mod_{g,n}(L);\mathbb{Q})$ stabilizes, then we have an isomorphism $H_k(Mod_{g,n}(L);\mathbb{Q}) \cong H_k(Mod_{g,n};\mathbb{Q})$. Indeed, since we are dealing with finite-index normal subgroups, the Hochschild-Serre spectral sequence collapses and gives that $H_k(Mod_{g,n};\mathbb{Q})$ is isomorphic to the co-invariants of the action of $Mod_{g,n}$ on $H_k(Mod_{g,n}(L);\mathbb{Q})$. However, stability implies that any Dehn twist acts trivially on $H_k(Mod_{g,n}(L);\mathbb{Q})$ (draw the picture -- the homology is entirely supported "away" from the simple closed curve), so we get the desired isomorphism.

Answer 2

This is not an answer to your question, but is directly related to your remark so I thought I should mention it.

I have recently proved, though I am afraid that it has not appeared yet, that moduli spaces of $r$-Spin curves exhibit homological stability. However, the truth of this statement depends sensitively on what one means by "moduli spaces of $r$-Spin curves":

If one takes the moduli stack $\mathcal{M}_{g}^{1/r}$ that represents families of Riemann surfaces equipped with a line bundle $\ell$ on the total space and a chosen isomorphism $\ell^{\otimes r} \cong \omega$ to the fibrewise cotangent bundle, then all is well and one has integral homology stability. However, if one takes the "rigidification" $\widetilde{\mathcal{M}}_{g}^{1/r}$ obtained by killing the natural $\mathbb{Z}/r$-worth of automorphisms of every object, the homology does not stabilise integrally, though it does over $\mathbb{Z}[1/r]$. In fact, even the first homology of $\widetilde{\mathcal{M}}_{g}^{1/r}$ jumps around all over the place.

The (orbifold) fundamental group of $\widetilde{\mathcal{M}}_{g}^{1/r}$ at some $r$-Spin surface $(\Sigma, \ell)$ may be identified with the subgroup $\widetilde{\Gamma}_g^{1/r} \leq \Gamma_g$ of those mapping classes which preserve the $r$-Spin structure $\ell$ up to isomorphism. Consequently, the groups $\widetilde{\Gamma}_g^{1/r}$ do not have integral stability. It is only a certain extension of these groups by $\mathbb{Z}/r$ which has integral stability.