Cohomology of mapping class groups and the abelian moduli space

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 $M = \Hom(\pi_1(\Sigma), U(1)) = \Hom(H_1(\Sigma),U(1))$, 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.