We prove a surgery formula of the Casson–Seiberg–Witten invariant of integral homology $S^1\times S^3$ 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 $S^1\times S^3$. 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].

中文翻译：

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积分同调S1×S3的Casson-Seiberg-Witten不变量的手术公式

我们证明了积分同调的 Casson-Seiberg-Witten 不变量的手术公式 ${秒}^1\times {秒}^3$ 沿着嵌入的环面，这可以被视为 Seiberg-Witten 不变量的乘积公式的扩展，或者是四维 Seiberg-Witten 同源理论中手术精确三角形的表现 ${秒}^1\times {秒}^3$. 作为一个应用，我们计算了这个不变量，用于在有限阶和不动点集的微分同胚下映射 3 流形的环面是一条简单的闭合曲线。该计算概括了 Lin-Ruberman-Saveliev 在 [On the monopole lefschetz number offinite order diffeomorphisms, Preprint, 2020] 中的结果。