# Splitting of the homology of the punctured mapping class group

## TL;DR

This paper proves a splitting of the homology of punctured mapping class groups, expressing it in terms of the homology of the unpunctured group and surface braid groups, with key computations of the latter's homology as a symplectic representation.

## Contribution

It establishes a homological splitting for punctured mapping class groups and computes the surface braid group's homology as a symplectic representation.

## Key findings

- Homology of punctured mapping class groups splits as a tensor product.
- Computed the homology of surface braid groups as symplectic representations.
- Provided explicit isomorphism relating these homologies.

## 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 $H_*(\Gamma_{g,1}^m;\mathbb{Z}_2)\cong H_*(\Gamma_{g,1};H_*(\beta_{m}(\Sigma_{g,1});\mathbb{Z}_2))$. The main ingredient is the computation of $H_*(\beta_{m}(\Sigma_{g,1});\mathbb{Z}_2)$ as a symplectic representation of $\Gamma_{g,1}$.