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

We consider the Birman-Hilden inclusion of the braid group into the mapping class group of an orientable surface with boundary, and prove that is stably trivial in homology with twisted coefficients in the symplectic representation 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 has only 4-torsion.