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

Abstract

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.