A surgery formula for the Casson–Seiberg–Witten invariant of integral homology S1×S3

We prove a surgery formula of the Casson–Seiberg–Witten invariant of integral homology S 1 × S 3 along an embedded torus, which could either be regarded as an extension of the product formula for Seiberg–Witten invariants or a manifestation of the surgery exact triangle in four-dimensional Seiberg–Witten theory of homology S 1 × S 3 . As an application, we compute this invariant for mapping tori of 3-manifolds under diffeomorphisms of finite-order and fixed-point set being a simple closed curve. This computation generalizes the result of Lin–Ruberman–Saveliev in [On the monopole lefschetz number of finite order diffeomorphisms, Preprint, 2020].