Abstract
Let be a knot in an integral homology –sphere and the corresponding –fold cyclic branched cover. Assuming that is a rational homology sphere (which is always the case when 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 . The proof relies on a careful analysis of the Seiberg–Witten equations on –orbifolds and of various –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 on integral homology –spheres; (2) we give a novel obstruction (in terms of the Jones polynomial) for the branched cover of a knot in being an –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.
Citation
Jianfeng Lin. Daniel Ruberman. Nikolai Saveliev. "On the monopole Lefschetz number of finite-order diffeomorphisms." Geom. Topol. 25 (7) 3591 - 3628, 2021. https://doi.org/10.2140/gt.2021.25.3591
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