#### Volume 25, issue 7 (2021)

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On the monopole Lefschetz number of finite-order diffeomorphisms

### Jianfeng Lin, Daniel Ruberman and Nikolai Saveliev

Geometry & Topology 25 (2021) 3591–3628
##### Abstract

Let $K$ be a knot in an integral homology $3$–sphere $Y$ and $\Sigma$ the corresponding $n$–fold cyclic branched cover. Assuming that $\Sigma$ is a rational homology sphere (which is always the case when $n$ is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of $\Sigma$. The proof relies on a careful analysis of the Seiberg–Witten equations on $3$–orbifolds and of various $\eta$–invariants. We give several applications of our formula: (1) we calculate the Seiberg–Witten and Furuta–Ohta invariants for the mapping tori of all semifree actions of $ℤ∕n$ on integral homology $3$–spheres; (2) we give a novel obstruction (in terms of the Jones polynomial) for the branched cover of a knot in ${S}^{3}$ being an $L$–space; and (3) we give a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers.

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##### Keywords
monopole, Seiberg–Witten, instantons, Floer homology, Furuta–Ohta invariant, $4$–manifold
##### Mathematical Subject Classification
Primary: 57R57
Secondary: 57K10, 57K31, 57K41, 57R58