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2011 Representation stability for the cohomology of the moduli space $\mathcal{M}_{g}^n$
Rita Jimenez Rolland
Algebr. Geom. Topol. 11(5): 3011-3041 (2011). DOI: 10.2140/agt.2011.11.3011


Let gn be the moduli space of Riemann surfaces of genus g with n labeled marked points. We prove that, for g2, 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 d3 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.


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Rita Jimenez Rolland. "Representation stability for the cohomology of the moduli space $\mathcal{M}_{g}^n$." Algebr. Geom. Topol. 11 (5) 3011 - 3041, 2011.


Received: 14 June 2011; Revised: 7 October 2011; Accepted: 8 October 2011; Published: 2011
First available in Project Euclid: 19 December 2017

zbMATH: 1269.55006
MathSciNet: MR2869450
Digital Object Identifier: 10.2140/agt.2011.11.3011

Primary: 55T05
Secondary: 57S05

Keywords: mapping class group , moduli space , representation stability

Rights: Copyright © 2011 Mathematical Sciences Publishers


Vol.11 • No. 5 • 2011
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