#### Volume 11, issue 5 (2011)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Representation stability for the cohomology of the moduli space $\mathcal{M}_{g}^n$

### Rita Jimenez Rolland

Algebraic & Geometric Topology 11 (2011) 3011–3041
##### Abstract

Let ${\mathsc{ℳ}}_{g}^{n}$ 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 ${\left\{{H}^{i}\left({\mathsc{ℳ}}_{g}^{n};ℚ\right)\right\}}_{n=1}^{\infty }$ 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 . We obtain representation stability for sequences , where is the mapping class group of many connected orientable manifolds $M$ of dimension $d\ge 3$ with centerless fundamental group; and for sequences , where is the classifying space of the subgroup of diffeomorphisms of $M$ that fix pointwise $n$ distinguished points in $M$.

##### Keywords
representation stability, moduli space, mapping class group
Primary: 55T05
Secondary: 57S05