A splitting theorem for the Seiberg-Witten invariant of a homology $S^1 \times S^3$

Abstract

We study the Seiberg–Witten invariant ${\lambda }_{SW}\left(X\right)$ of smooth spin $4$–manifolds $X$ with the rational 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 Frøyshov invariant $h\left(X\right)$ and a certain Lefschetz number in the reduced monopole Floer homology of Kronheimer and Mrowka. We apply this formula to obstruct the existence of metrics of positive scalar curvature on certain $4$–manifolds, and to exhibit new classes of homology $3$–spheres of infinite order in the homology cobordism group.