The Seiberg-Witten equations on end-periodic manifolds and an obstruction to positive scalar curvature metrics

By studying the Seiberg-Witten equations on end-periodic manifolds, we give an obstruction on the existence of positive scalar curvature metric on compact $4$-manifolds with the same homology as $S^1\times S^3$. This obstruction is given in terms of the relation between the Fr{\o}yshov invariant of the generator of $H_3(X;Z)$ with the $4$-dimensional Casson invariant ${\lambda }_{SW}(X)$ defined by Mrowka-Ruberman-Saveliev. Along the way, we develop a framework that can be useful in further study of the Seiberg-Witten theory on general end-periodic manifolds.