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 analogue of the additivity of the Casson invariant under connected-sum and splicing along knots.