Andrea Bianchi
Published 2019-03-14Version 1
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 $H_*(\Gamma_{g,1}^m;\mathbb{Z}_2)\cong H_*(\Gamma_{g,1};H_*(\beta_{m}(\Sigma_{g,1});\mathbb{Z}_2))$. The main ingredient is the computation of $H_*(\beta_{m}(\Sigma_{g,1});\mathbb{Z}_2)$ as a symplectic representation of $\Gamma_{g,1}$.
Comments: 30 pages, 16 figures
Braid groups, mapping class groups and their homology with twisted coefficients
The cohomology of the braid group B_3 and of SL_2(Z) with coefficients in a geometric representation
Representations of braid groups via conjugation actions on congruence subgroups
![QR code for arXiv:1903.06065 [math.AT]](http://oss.arxitics.com/images/favicon.svg)