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

## TL;DR

This paper establishes a splitting formula for the Seiberg-Witten invariant of certain 4-manifolds, linking it to the Fr{\

## Contribution

It introduces a splitting formula for the Seiberg-Witten invariant in terms of monopole Floer homology and the Fr{\

## Key findings

- Derived a splitting formula connecting the Seiberg-Witten invariant with Floer homology.
- Obstructed the existence of positive scalar curvature metrics on specific 4-manifolds.
- Identified new homology 3-spheres with Rohlin invariant one and infinite order in cobordism group.

## Abstract

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.