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 $M = \Hom(π_1(Σ), U(1)) = \Hom(H_1(Σ),U(1))$, and hence both $L^2(M)$ and $C^{\infty }(M)$ are modules over $\Gamma$. In this paper, we prove that both the cohomology groups $H^1(\Gamma ,L^2(M))$ and $H^1(\Gamma ,C^{\infty }(M))$ vanish.

18 pages, 3 figures

References in corpus (2)

• Mapping Class Groups do not have Kazhdan's Property (T)
• Hitchin's connection, Toeplitz operators and symmetry invariant deformation quantization

Cited by in corpus (1)

• Asymptotics of Toeplitz operators and applications in TQFT