JournalsqtVol. 3 , No. 3/4DOI 10.4171/qt/32

Cohomology of mapping class groups and the abelian moduli space

  • Jørgen Ellegaard Andersen

    Aarhus University, Denmark
  • Rasmus Villemoes

    Aarhus University, Denmark
Cohomology of mapping class groups and the abelian moduli space cover

Abstract

We consider a surface Σ\Sigma of genus g3g \geq 3, either closed or with exactly one puncture. The mapping class group Γ\Gamma of Σ\Sigma acts symplectically on the abelian moduli space M=Hom(π1(Σ),U(1))=Hom(H1(Σ),U(1))M = \operatorname{Hom}(\pi_1(\Sigma), \operatorname{U}(1)) = \operatorname{Hom}(H_1(\Sigma), \operatorname{U}(1)), and hence both L2(M)L^2(M) and C(M)C^\infty(M) are modules over Γ\Gamma. In this paper, we prove that both the cohomology groups H1(Γ,L2(M))H^1(\Gamma, L^2(M)) and H1(Γ,C(M))H^1(\Gamma, C^\infty(M)) vanish.