2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/224461We consider a surface $Σ$ of genus $g \geq 3$, either closed or with exactly one puncture. The mapping class group $Γ$ of $Σ$ acts symplectically on the abelian moduli space $M = \Hom(π_1(Σ), U(1)) = \Hom(H_1(Σ),U(1))$, and hence both $L^2(M)$ and $C^\infty(M)$ are modules over $Γ$. In this paper, we prove that both the cohomology groups $H^1(Γ, L^2(M))$ and $H^1(Γ, C^\infty(M))$ vanish.18 pages, 3 figuresDifferential Geometrytext