A Splitting Theorem For The Seiberg-Witten Invariant Of A Homology S-1 X S-3

We study the Seiberg-Witten invariant lambda(SW)(X) of smooth spin 4-manifolds X with the rational homology of S-1 x 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 Froyshov 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.