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 >= 2, the cohomology groups {H-i(M-g(n); Q)}(n=1)(infinity) form a sequence of S-n-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(g)(n). We obtain representation stability for sequences {H-i(PMod(n)(M); Q)}(n=1)(infinity), 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(BPDiff(n)(M); Q)}(n=1)(infinity), where BPDiff(n) (M) is the classifying space of the sub-group PDiff(n)(M) of diffeomorphisms of M that fix pointwise n distinguished points in M.