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Splitting of the homology of the punctured mapping class group
Journal of Topology  (IF1.582),  Pub Date : 2020-05-20, DOI: 10.1112/topo.12153
Andrea Bianchi

Let $Γ g , 1 m$ be the mapping class group of the orientable surface $Σ g , 1 m$ of genus $g$ with one parametrized 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 $H ∗ ( Γ g , 1 m ; Z 2 ) ≅ H ∗ ( Γ g , 1 ; H ∗ ( β m ( Σ g , 1 ) ; Z 2 ) )$. The main ingredient is the computation of $H ∗ ( β m ( Σ g , 1 ) ; Z 2 )$ as a symplectic representation of $Γ g , 1$.