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

TL;DR

This paper introduces the $ ext{Pin}^-(2)$-monopole invariant for 4-manifolds, computes it for various examples, and uses it to construct exotic smooth structures and estimate surface genus in 4-manifolds.

Contribution

It defines a new diffeomorphism invariant based on $ ext{Pin}^-(2)$-monopole equations, with applications to exotic smooth structures and genus bounds.

Findings

Computed invariants for several 4-manifolds.

Proved gluing formulae for the invariants.

Constructed exotic smooth structures on connected sums.

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 $X$.