  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.
