Representation stability for the cohomology of the moduli space M ng

Abstract

Let M ng be the moduli space of Riemann surfaces of genus g with n labeled marked points. We prove that, for g ≥ 2, the cohomology groups {H i (M ng;Q} ∞n = 1 form a sequence of Sn -representations which is representation stable in the sense of Church-Farb [7]. In particular this result applied to the trivial S n -representation implies rational "puncture homological stability" for the mapping class group Mod ng. We obtain representation stability for sequences {H i (PMod n (M); Q} ∞n = 1, where PMod n(M) is the mapping class group of many connected orientable manifolds M of dimension d ≥ 3 with centerless fundamental group; and for sequences {H i(PMod n(M); Q} ∞n = 1, where BPDiff n(M) is the classifying space of the subgroup PDiff n(M) of diffeomorphisms of M that fix pointwise n distinguished points in M.