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arXiv · 0903.4045

Cohomology of mapping class groups and the abelian moduli space

Jørgen Ellegaard Andersen·Rasmus Villemoes
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Abstract

We 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.

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Jørgen Ellegaard Andersen, Rasmus Villemoes. 2009-03-24. Cohomology of mapping class groups and the abelian moduli space. https://arxiv.org/abs/0903.4045

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