Representation stability for the cohomology of the moduli space $\mathcal{M}_{g}^n$

Abstract

Let ${\mathcal{ℳ}}_g^n$ be the moduli space of Riemann surfaces of genus $g$ with $n$ labeled marked points. We prove that, for $g\ge 2$, the cohomology groups ${\left\{H^i\left({\mathcal{ℳ}}_g^n;\mathbb{Q}\right)\right\}}_{n=1}^{\infty }$ form a sequence of $S_n$–representations which is representation stable in the sense of Church–Farb. In particular this result applied to the trivial $S_n$–representation implies rational “puncture homological stability” for the mapping class group ${\text{ Mod}}_g^n$. We obtain representation stability for sequences ${\left\{H^i\left({\text{ PMod}}^n\left(M\right);\mathbb{Q}\right)\right\}}_{n=1}^{\infty }$, where ${\text{ PMod}}^n\left(M\right)$ is the mapping class group of many connected orientable manifolds $M$ of dimension $d\ge 3$ with centerless fundamental group; and for sequences ${\left\{H^i\left(\right.B{\text{ PDiff}}^n\left(M\right);\mathbb{Q}\left)\right.\right\}}_{n=1}^{\infty }$, where $B{\text{ PDiff}}^n\left(M\right)$ is the classifying space of the subgroup ${\text{ PDiff}}^n\left(M\right)$ of diffeomorphisms of $M$ that fix pointwise $n$ distinguished points in $M$.