On the monopole Lefschetz number of finite-order diffeomorphisms

Lin, Jianfeng¹ ; Ruberman, Daniel² ; Saveliev, Nikolai³

¹ Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
² Department of Mathematics, Brandeis University, Waltham, MA, United States
³ Department of Mathematics, University of Miami, Coral Gables, FL, United States

Geometry & topology, Tome 25 (2021) no. 7, pp. 3591-3628.

Résumé

Let $K$ be a knot in an integral homology $3$–sphere $Y$ and $\Sigma$ the corresponding $n$–fold cyclic branched cover. Assuming that $\Sigma$ 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 $\Sigma$. The proof relies on a careful analysis of the Seiberg–Witten equations on $3$–orbifolds and of various $\eta$–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 $\mathbb{Z}∕n$ 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.

Détail

DOI : 10.2140/gt.2021.25.3591

Keywords: monopole, Seiberg–Witten, instantons, Floer homology, Furuta–Ohta invariant, $4$–manifold

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