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.