# Cohomology of mapping class groups and the abelian moduli space

### Jørgen Ellegaard Andersen

Aarhus University, Denmark### Rasmus Villemoes

Aarhus University, Denmark

## Abstract

We consider a surface $\Sigma$ of genus $g \geq 3$, either closed or with exactly one puncture. The mapping class group $\Gamma$ of $\Sigma$ acts symplectically on the abelian moduli space $M = \operatorname{Hom}(\pi_1(\Sigma), \operatorname{U}(1)) = \operatorname{Hom}(H_1(\Sigma), \operatorname{U}(1))$, and hence both $L^2(M)$ and $C^\infty(M)$ are modules over $\Gamma$. In this paper, we prove that both the cohomology groups $H^1(\Gamma, L^2(M))$ and $H^1(\Gamma, C^\infty(M))$ vanish.