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European Mathematical Society Publishing House
2016-09-19 17:05:39
Quantum Topology
Quantum Topol.
QT
1663-487X
1664-073X
General
10.4171/QT
http://www.ems-ph.org/doi/10.4171/QT
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
3
2012
3
Cohomology of mapping class groups and the abelian moduli space
Jørgen Ellegaard
Andersen
University of Aarhus, AARHUS C, DENMARK
Rasmus
Villemoes
University of Aarhus, AARHUS, DENMARK
Mapping class groups, group cohomology, moduli space, property (T)
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.
Manifolds and cell complexes
Group theory and generalizations
Differential geometry
General
359
376
10.4171/QT/32
http://www.ems-ph.org/doi/10.4171/QT/32