Surgery formulas for Seiberg-Witten invariants and family Seiberg-Witten invariants

TL;DR

This paper develops surgery formulas for Seiberg-Witten invariants and their family versions, enabling analysis of how surgeries affect the invariants and can lead to exotic smooth structures on 4-manifolds.

Contribution

It introduces new surgery formulas for both ordinary and family Seiberg-Witten invariants, linking post-surgery invariants to the original moduli space and cohomology classes.

Findings

Formulas express post-surgery invariants in terms of original moduli spaces.

Surgery can preserve or create exotic smooth structures.

Applications to understanding smooth structures on 4-manifolds.

Abstract

We prove a surgery formula for the ordinary Seiberg-Witten invariants, and surgery formulas for the families Seiberg-Witten invariants of families of $4$-manifolds obtained through fibrewise surgery. 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. We use these surgery formulas to study how a surgery can preserve or produce exotic phenomena.