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# Mathematical Research Letters

## Volume 29 (2022)

### Number 4

### Fiber sum formulae for the Casson–Seiberg–Witten invariant of integral homology $S^1 \times S^3$

Pages: 1197 – 1227

DOI: https://dx.doi.org/10.4310/MRL.2022.v29.n4.a11

#### Author

#### 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 analog of the additivity of the Casson invariant under connected-sum and splicing along knots. As an application, we compute the Casson–Seiberg–Witten invariant of a family of integral homology $S^1 \times S^3$ which arises as the mapping tori under maps of infinite order in the mapping class group of certain $3$‑manifolds.

Received 14 February 2020

Accepted 23 June 2020

Published 23 February 2023