Representation stability for the cohomology of the moduli space M_g^n

Abstract

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 Sn representations which is representation stable in the sense of Church-Farb [CF]. In particular this result applied to the trivial Sn 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 manifolds M of dimension d \geq 3 with centerless fundamental group; and for sequences {H^i(BPDiff^n(M);Q)}_{n=1}^{\infty}, 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.