On signatures and a subgroup of a central extension to the mapping class group

Atiyah’s work [1] describes the relationship between multiplication in a central extension of the mapping class group of a surface of genus $n$ and the signatures of $4$-dimensional manifolds. This work studies a subgroup of the central extension, which comes from the image of a representation of the pure framed braid group on $n$-strands found in [5], and the signatures of corresponding $4$-manifolds via a split exact sequence. We construct a splitting map to prove the sequence is split exact, and we use the splitting to give a topological description of homology classes in $4$-dimensional manifolds with non-zero intersection. We conclude with a description of multiplication in the subgroup.

Keywords

mapping class group, pure braids, signatures

2010 Mathematics Subject Classification

20F36, 57M25, 57N05, 57N13

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Published 1 January 2003