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

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 semifree actions of 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; 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.

monopole, Seiberg–Witten, instantons, Floer homology, Furuta–Ohta invariant, $4$–manifold
Mathematical Subject Classification
Primary: 57R57
Secondary: 57K10, 57K31, 57K41, 57R58
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
Jianfeng Lin
Yau Mathematical Sciences Center
Tsinghua University
Daniel Ruberman
Department of Mathematics
Brandeis University
Waltham, MA
United States
Nikolai Saveliev
Department of Mathematics
University of Miami
Coral Gables, FL
United States