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

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.

Full Text (PDF format)

Received 14 February 2020

Accepted 23 June 2020

Published 23 February 2023