A surgery formula for Seiberg-Witten invariants

TL;DR

This paper establishes a surgery formula for Seiberg-Witten invariants of 4-manifolds with first Betti number one, enabling the study of exotic smooth structures and genus bounds in nonsimply connected cases.

Contribution

It provides a new surgery formula for Seiberg-Witten invariants in the case of 4-manifolds with b_1=1, extending the toolkit for analyzing smooth structures.

Findings

Formula relates post-surgery invariants to original moduli space

Enables detection of exotic smooth structures

Provides genus bounds for embedded surfaces

Abstract

We prove a surgery formula for the ordinary Seiberg-Witten invariants of smooth $4$-manifolds with $b_1 =1$. Our formula expresses the Seiberg-Witten invariants of the manifold after the surgery, in terms of the original Seiberg-Witten moduli space cut down by a cohomology class in the configuration space. This formula can be used to find exotic smooth structures on nonsimply connected $4$-manifolds, and gives a lower bound of the genus of an embedding surface in nonsimply connected $4$-manifolds. In forthcoming work, we will extend these results to give a surgery formula for the families Seiberg-Witten invariants.