A splitting theorem for the Seiberg-Witten invariant of a homology S1× S3

Jianfeng Lin, Daniel Ruberman, Nikolai Saveliev
2018 Geometry and Topology  
We study the Seiberg-Witten invariant $\lambda_{\rm{SW}} (X)$ of smooth spin $4$-manifolds $X$ with integral homology of $S^1\times S^3$ defined by Mrowka, Ruberman, and Saveliev as a signed count of irreducible monopoles amended by an index-theoretic correction term. We prove a splitting formula for this invariant in terms of the Fr{\o}yshov invariant $h(X)$ and a certain Lefschetz number in the reduced monopole Floer homology of Kronheimer and Mrowka. We apply this formula to obstruct
more » ... to obstruct existence of metrics of positive scalar curvature on certain 4-manifolds, and to exhibit new classes of integral homology $3$-spheres of Rohlin invariant one which have infinite order in the homology cobordism group.
doi:10.2140/gt.2018.22.2865 fatcat:e3skqqb5lvd4hgl2e6fvbhyuxe