On the monopole Lefschetz number of finite-order diffeomorphisms

Lin, Jianfeng; Ruberman, Daniel; Saveliev, Nikolai

doi:10.2140/gt.2021.25.3591

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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

DOI: 10.2140/gt.2021.25.3591

Abstract

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 $ℤ ∕ n$ 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

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

Authors

Jianfeng Lin
Yau Mathematical Sciences Center Tsinghua University Beijing China

Daniel Ruberman
Department of Mathematics Brandeis University Waltham, MA United States

Nikolai Saveliev
Department of Mathematics University of Miami Coral Gables, FL United States

Volume 25, issue 7 (2021)