Fiber Sum Formulae for the Casson-Seiberg-Witten Invariant of Integral Homology $S^1 \times S^3$

TL;DR

This paper proves that the Casson-Seiberg-Witten invariant for certain 4-manifolds is additive under fiber sum operations along embedded curves and tori, extending known 3-dimensional invariants to a 4-dimensional context.

Contribution

It establishes the additivity property of the Casson-Seiberg-Witten invariant under fiber sum along embedded curves and tori in integral homology S^1 x S^3.

Findings

Proves additivity of the invariant under fiber sum operations.

Extends the analogy of Casson invariant properties from 3 to 4 dimensions.

Provides a new tool for studying 4-manifolds with similar structures.

Abstract

We prove the additivity of the Casson-Seiberg-Witten invariant of integral homology $S^1 \times S^3$ under fiber sum along embedded curves and embedded tori, which is the $4$-dimensional analogue of the additivity of the Casson invariant under connected-sum and splicing along knots.