Is there Harer stability for moduli of curves with level structure? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:40:15Z http://mathoverflow.net/feeds/question/52406 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52406/is-there-harer-stability-for-moduli-of-curves-with-level-structure Is there Harer stability for moduli of curves with level structure? Dan Petersen 2011-01-18T13:54:18Z 2011-01-18T14:58:31Z <p>The famous Harer stability theorem asserts that the homology group $H_d(\mathcal{M}_{g,n},\mathbf{Z})$ is independent of <em>g</em> and <em>n</em> in the range $0 \leq 2d &lt; g-1$. 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). </p> <p>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 <em>G</em>, and denote by $\mathcal{M}_{g,n}[G]$ the moduli space parametrizing <em>n</em>-pointed smooth curves of genus <em>g</em> equipped with an étale <em>G</em>-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.</p> <p>Remark: The corresponding statement for moduli of <em>spin curves</em> 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 <em>r</em>-spin curves and the spaces of curves with $G=\mathbf{Z}/r\mathbf{Z}$ level structure. </p> http://mathoverflow.net/questions/52406/is-there-harer-stability-for-moduli-of-curves-with-level-structure/52407#52407 Answer by Andy Putman for Is there Harer stability for moduli of curves with level structure? Andy Putman 2011-01-18T14:15:00Z 2011-01-18T14:15:00Z <p>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).</p> <p>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".</p> <p>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.</p> http://mathoverflow.net/questions/52406/is-there-harer-stability-for-moduli-of-curves-with-level-structure/52409#52409 Answer by Oscar Randal-Williams for Is there Harer stability for moduli of curves with level structure? Oscar Randal-Williams 2011-01-18T14:58:31Z 2011-01-18T14:58:31Z <p>This is not an answer to your question, but is directly related to your remark so I thought I should mention it.</p> <p>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":</p> <p>If one takes the moduli stack <code>$\mathcal{M}_{g}^{1/r}$</code> 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" <code>$\widetilde{\mathcal{M}}_{g}^{1/r}$</code> 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 <code>$\widetilde{\mathcal{M}}_{g}^{1/r}$</code> jumps around all over the place.</p> <p>The (orbifold) fundamental group of <code>$\widetilde{\mathcal{M}}_{g}^{1/r}$</code> 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 $\widetilde{\Gamma}_g^{1/r}$ <em>do not have integral stability</em>. It is only a certain extension of these groups by $\mathbb{Z}/r$ which has integral stability.</p>