Quantum Topology


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Volume 3, Issue 3/4, 2012, pp. 359–376
DOI: 10.4171/QT/32

Published online: 2012-05-30

Cohomology of mapping class groups and the abelian moduli space

Jørgen Ellegaard Andersen[1] and Rasmus Villemoes[2]

(1) Aarhus University, Denmark
(2) Aarhus University, Denmark

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.

Keywords: Mapping class groups, group cohomology, moduli space, property (T)

Andersen Jørgen Ellegaard, Villemoes Rasmus: Cohomology of mapping class groups and the abelian moduli space. Quantum Topol. 3 (2012), 359-376. doi: 10.4171/QT/32