$$\text{ Pin }^-(2)$$ -monopole equations and intersection forms with local coefficients of four-manifolds

Abstract

We introduce a variant of the Seiberg-Witten equations, \(\text{ Pin }^-(2)\)-monopole equations, and give its applications to intersection forms with local coefficients of four-manifolds. The first application is an analogue of Froyshov’s results on four-manifolds with definite intersection forms with local coefficients. The second is a local coefficient version of Furuta’s \(10/8\)-inequality. As a corollary, we construct nonsmoothable spin four-manifolds satisfying Rohlin’s theorem and the \(10/8\)-inequality.