Self-Gluing formula of the monopole invariant and its application

TL;DR

This paper establishes a gluing formula relating the Seiberg-Witten invariants of a 4-manifold before and after removing neighborhoods of embedded surfaces and gluing boundaries, with applications to symplectic structures.

Contribution

It introduces a novel self-gluing formula for monopole invariants and demonstrates its use in analyzing symplectic structures on 4-manifolds.

Findings

Derived a formula linking Seiberg-Witten invariants of original and modified 4-manifolds.

Showed the formula's application in determining the existence of symplectic structures.

Provided new tools for studying 4-manifold topology through monopole invariants.

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.