A splitting theorem for the seiberg-witten invariant of a homology S1 × S3

Abstract

We study the Seiberg-Witten invariant λSW(X) of smooth spin 4 -manifolds X with the rational homology of S1 × S3 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(X) 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.