# Braid groups, mapping class groups and their homology with twisted coefficients

## TL;DR

This paper proves that the homology of a certain braid group inclusion into a mapping class group becomes trivial with twisted coefficients in the stable range, and reveals that the stable homology has only 4-torsion.

## Contribution

It generalizes previous results by showing stable triviality of the homology with twisted coefficients and identifies the torsion nature of the stable homology.

## Key findings

- Homology with twisted coefficients is stably trivial.
- Stable homology contains only 4-torsion.
- Generalizes results from constant to twisted coefficients.

## Abstract

We consider the Birman-Hilden inclusion $\varphi\colon\mathfrak{Br}_{2g+1}\to\Gamma_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $\varphi$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(\Sigma_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in $\varphi^*(H_1(\Sigma_{g,1}))$ has only $4$-torsion.