Cohomology of mapping class groups and the abelian moduli space

Details Cohomology of mapping class groups and the abelian moduli space Cohomology of mapping class groups and the abelian moduli space

2009-03-24

arxiv.org/abs/0903.4045v1

Mathematics

Differential Geometry

18 pages, 3 figures

Scientific paper

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 $M = \Hom(\pi_1(\Sigma), U(1)) = \Hom(H_1(\Sigma),U(1))$, 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.

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