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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
{Hi BPDiffn(M); Q }n=1∞, where BPDiffn(M) is the classifying space of the
subgroup PDiffn(M) of diffeomorphisms of M that fix pointwise n distinguished
points in M.
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