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On
the monopole Lefschetz number of finite-order
diffeomorphisms
Jianfeng Lin, Daniel Ruberman and Nikolai Saveliev |
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Geometry & Topology 25 (2021) 3591–3628
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Abstract |
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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. |
Keywords
monopole, Seiberg–Witten, instantons, Floer homology,
Furuta–Ohta invariant, $4$–manifold
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Mathematical Subject Classification
Primary: 57R57
Secondary: 57K10, 57K31, 57K41, 57R58
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References |
Publication
Received: 11 May 2020
Revised: 18 December 2020
Accepted: 23 December 2020
Published: 25 January 2022
Proposed: Simon Donaldson
Seconded: Ciprian Manolescu, András I Stipsicz
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Authors |
| Jianfeng Lin | |
| Yau Mathematical Sciences
Center Tsinghua University Beijing China |
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| Daniel Ruberman | |
| Department of Mathematics Brandeis University Waltham, MA United States |
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| Nikolai Saveliev | |
| Department of Mathematics University of Miami Coral Gables, FL United States |
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