On the monopole Lefschetz number of finite order diffeomorphisms

林剑锋 清华大学丘成桐数学科学中心 Daniel Ruberman Brandeis University Nikolai Saveliev University of Miami

Geometric Analysis and Geometric Topology mathscidoc:2205.15005

Geometry & Topology, 25, 2021.7
Let K be a knot in an integral homology 3-sphere Y, and Σ the corresponding n-fold cyclic branched cover. Assuming that Σ 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 Σ. The proof relies on a careful analysis of the Seiberg--Witten equations on 3-orbifolds and of various η-invariants. We give several applications of our formula: (1) we calculate the Seiberg--Witten and Furuta--Ohta invariants for the mapping tori of all semi-free 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 S3 being an L-space; (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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  title={On the monopole Lefschetz number of finite order diffeomorphisms},
  author={林剑锋, Daniel Ruberman, and Nikolai Saveliev},
  booktitle={Geometry & Topology},
林剑锋, Daniel Ruberman, and Nikolai Saveliev. On the monopole Lefschetz number of finite order diffeomorphisms. 2021. Vol. 25. In Geometry & Topology. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517174430807766250.
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