Representation stability for the cohomology of the moduli space ℳgn

Jimenez Rolland, Rita¹

¹Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago IL 60637, USA

Algebraic and Geometric Topology, Tome 11 (2011) no. 5, pp. 3011-3041

Résumé

Let ℳgn 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(ℳgn; ℚ)}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); ℚ)}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); ℚ}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.

Détail

DOI : 10.2140/agt.2011.11.3011

Keywords: representation stability, moduli space, mapping class group

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