Cohomology of mapping class groups and the abelian moduli space

Abstract

We consider a surface $\Sigma$ of genus $g\ge 3$, either closed or with exactly one puncture. The mapping class group $\Gamma$ of $\Sigma$ acts symplectically on the abelian moduli space $M=\mathrm{Hom}({\pi }_1(\Sigma ),U(1))=\mathrm{Hom}(H_1(\Sigma ),U(1))$, 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.