映射类群的上同调与交换模空间
Cohomology of mapping class groups and the abelian moduli space
摘要：
我们考虑一个表面$Σ$属$g \geq 3$，无论是封闭的或只有一个穿刺。$Σ$的映射类群$Γ$辛地作用在交换模空间$M = \Hom(π_1(Σ), U(1)) = \Hom(H_1(Σ),U(1))$上，因此$L^2(M)$和$C^\infty(M)$都是$Γ$上的模。本文证明了上同调群$H^1(Γ, L^2(M))$和$H^1(Γ, C^\infty(M))$均为零.
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
