  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 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.
