Volume 11, issue 5 (2011)

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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
Publication
Received: 14 June 2011
Revised: 7 October 2011
Accepted: 8 October 2011
Published: 14 December 2011
Authors
 Rita Jimenez Rolland Department of Mathematics University of Chicago 5734 University Avenue Chicago IL 60637 USA http://www.math.uchicago.edu/~atir83/