$\mathrm{Pin}^{-} (2)$-monopole invariants

Abstract

We introduce a diffeomorphism invariant of 4-manifolds, the $\mathrm{Pin}^{-} (2)$-monopole invariant, defined by using the $\mathrm{Pin}^{-} (2)$-monopole equations. We compute the invariants of several 4-manifolds, and prove gluing formulae. By using the invariants, we construct exotic smooth structures on the connected sum of an elliptic surface $E(n)$ with arbitrary number of the 4-manifolds of the form of $S^2 \times \Sigma$ or $S^1 \times Y$ where $\Sigma$ is a compact Riemann surface with positive genus and $Y$ is a closed 3-manifold. As another application, we give an estimate of the genus of surfaces embedded in a 4-manifold $X$ representing a class $\alpha \in H_2 (X; l)$, where $l$ is a local coefficient on $X$.