Splitting of the homology of the punctured mapping class group

Abstract

Let ${\Gamma }_{g,1}^m$ be the mapping class group of the orientable surface ${\Sigma }_{g,1}^m$ of genus $g$ with one parametrised boundary curve and $m$ permutable punctures; when $m=0$ we omit it from the notation. Let ${\beta }_m({\Sigma }_{g,1})$ be the braid group on $m$ strands of the surface ${\Sigma }_{g,1}$. We prove that $H_{\ast }({\Gamma }_{g,1}^m;{\mathbb{Z}}_2)\cong H_{\ast }({\Gamma }_{g,1};H_{\ast }({\beta }_m({\Sigma }_{g,1});{\mathbb{Z}}_2))$. The main ingredient is the computation of $H_{\ast }({\beta }_m({\Sigma }_{g,1});{\mathbb{Z}}_2)$ as a symplectic representation of ${\Gamma }_{g,1}$.

32 pages, 16 figures. Accepted for publication in Journal of Topology