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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

Let gn be the moduli space of Riemann surfaces of genus g with n labeled marked points. We prove that, for g 2, the cohomology groups {Hi(gn; )}n=1 form a sequence of Sn–representations which is representation stable in the sense of Church–Farb. In particular this result applied to the trivial Sn–representation implies rational “puncture homological stability” for the mapping class group  Modgn. We obtain representation stability for sequences {Hi( PModn(M); )}n=1, where  PModn(M) is the mapping class group of many connected orientable manifolds M of dimension d 3 with centerless fundamental group; and for sequences {HiB PDiffn(M); }n=1, where B PDiffn(M) is the classifying space of the subgroup  PDiffn(M) of diffeomorphisms of M that fix pointwise n distinguished points in M.

representation stability, moduli space, mapping class group
Mathematical Subject Classification 2000
Primary: 55T05
Secondary: 57S05
Received: 14 June 2011
Revised: 7 October 2011
Accepted: 8 October 2011
Published: 14 December 2011
Rita Jimenez Rolland
Department of Mathematics
University of Chicago
5734 University Avenue
Chicago IL 60637