Self-Gluing formula of the monopole invariant and its application

Abstract

Given a $4$-manifold $\hat{M}$ and two homeomorphic surfaces ${\Sigma }_1,{\Sigma }_2$ smoothly embedded in $\hat{M}$ with genus more than 1, we remove the neighborhoods of the surfaces and obtain a new $4$-manifold $M$ from gluing two boundaries $S^1\times {\Sigma }_1$ and $S^1\times {\Sigma }_1.$ In this artice, we prove a gluing formula which describes the relation of the Seiberg-Witten invariants of $M$ and $\hat{M}.$ Moreover, we show the application of the formula on the existence condition of the symplectic structure on a family of $4$-manfolds under some conditions.