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Braid groups, mapping class groups and their homology with twisted coefficients (1901.08028v2)
Published 23 Jan 2019 in math.AT
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.