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

Abstract

We study the Seiberg–Witten invariant [math] of smooth spin [math] –manifolds [math] with the rational homology of [math] 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 [math] 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 [math] –manifolds, and to exhibit new classes of homology [math] –spheres of infinite order in the homology cobordism group.