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arXiv · 1903.06065

Splitting of the homology of the punctured mapping class group

Andrea Bianchi
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Abstract

Let $Γ_{g,1}^m$ be the mapping class group of the orientable surface $Σ_{g,1}^m$ of genus $g$ with one parametrised boundary curve and $m$ permutable punctures; when $m=0$ we omit it from the notation. Let $β_{m}(Σ_{g,1})$ be the braid group on $m$ strands of the surface $Σ_{g,1}$. We prove that $H_*(Γ_{g,1}^m;\mathbb{Z}_2)\cong H_*(Γ_{g,1};H_*(β_{m}(Σ_{g,1});\mathbb{Z}_2))$. The main ingredient is the computation of $H_*(β_{m}(Σ_{g,1});\mathbb{Z}_2)$ as a symplectic representation of $Γ_{g,1}$.

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Andrea Bianchi. 2020-05-01. Splitting of the homology of the punctured mapping class group. https://doi.org/10.1112/topo.12153

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