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

TL;DR

This paper proves that the cohomology groups of the moduli space of Riemann surfaces with marked points exhibit representation stability, leading to homological stability results for related mapping class groups and diffeomorphism groups.

Contribution

It establishes representation stability for the cohomology of moduli spaces and related groups, extending stability results to higher-dimensional manifolds and their diffeomorphism groups.

Findings

Cohomology groups form representation stable sequences for g ≥ 2.

Rational puncture homological stability for the mapping class group.

Representation stability for mapping class groups of higher-dimensional manifolds.

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.