On the stable cohomology algebra of extended mapping class groups for surfaces

Abstract

Let $\Sigma_{g,1}$ be an oriented compact surface of genus $g$ with 1 boundary component, and $\Gamma_{g,1}$ the mapping class group of $\Sigma_{g,1}$. We determine the stable cohomology group of $\Gamma_{g,1}$ with coefficients in $H^1 (\Sigma_{g ,1} ; \mathbb{Z})^{\otimes n}$, $n \ge 1$, explicitly modulo the stable cohomology group with trivial coefficients. As a corollary the rational stable cohomology algebra of the semi-direct product $\Gamma_{g,1} \ltimes H_1 (\Sigma_{g,1} ; \mathbb{Z})$ (which we call the extended mapping class group) is proved to be freely generated by the generalized Morita-Mumford classes $\widetilde{m_{i,j}}$'s $(i \ge 0,\, j \ge 1,\, i+j \ge 2)$ [11] over the rational stable cohomology algebra of the group $\Gamma_{g,1}$.