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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 .