On the monopole Lefschetz number of finite order diffeomorphisms

Abstract

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 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 $S^3$ 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.

arXiv comment39 page, 2 figures. Added a reference to Langte Ma's paper arXiv:1909.01533, which contains an independent proof of our Theorem B. Final version, to appear in Geometry and Topology

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Cited by in corpus (6)

• Rank inequalities for the Heegaard Floer homology of branched covers
• Links all of whose cyclic branched covers are L-spaces
• Involutions, knots, and Floer K-theory
• A monopole invariant for families of contact structures
• On the spectral sets of Inoue surfaces
• Surgery and Excision for Furuta-Ohta invariants on Homology $S^1\times S^3$