Representation stability for the cohomology of the moduli space ℳgn

Let $ℳ_{g}^{n}$ be the moduli space of Riemann surfaces of genus $g$ with $n$ labeled marked points. We prove that, for $g \geq 2$ , the cohomology groups ${H^{i} (ℳ_{g}^{n}; ℚ)}_{n = 1}^{\infty}$ 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 ${Mod}_{g}^{n}$ . We obtain representation stability for sequences ${H^{i} ({PMod}^{n} (M); ℚ)}_{n = 1}^{\infty}$ , where ${PMod}^{n} (M)$ is the mapping class group of many connected orientable manifolds $M$ of dimension $d \geq 3$ with centerless fundamental group; and for sequences ${H^{i} (B {PDiff}^{n} (M); ℚ)}_{n = 1}^{\infty}$ , where $B {PDiff}^{n} (M)$ is the classifying space of the subgroup ${PDiff}^{n} (M)$ of diffeomorphisms of $M$ that fix pointwise $n$ distinguished points in $M$ .

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Keywords

representation stability, moduli space, mapping class group

Mathematical Subject Classification 2000

Primary: 55T05

Secondary: 57S05

References

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/

Volume 11, issue 5 (2011)

Rita Jimenez Rolland

Abstract