Cohomology of mapping class groups and the abelian moduli space

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.