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arXiv · 1901.08028

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

Andrea Bianchi
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Abstract

We consider the Birman-Hilden inclusion $φ\colon\mathfrak{Br}_{2g+1}\toΓ_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $φ$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(Σ_{g,1})$ 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 $φ^*(H_1(Σ_{g,1}))$ has only $4$-torsion.

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Andrea Bianchi. 2019-01-27. Braid groups, mapping class groups and their homology with twisted coefficients. https://arxiv.org/abs/1901.08028

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