On the monopole Lefschetz number of finite-order diffeomorphisms

Abstract

Let K be a knot in an integral homology 3–sphere Y and † the corresponding n–fold cyclic branched cover. Assuming that † 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 †. The proof relies on a careful analysis of the Seiberg–Witten equations on 3–orbifolds and of various n–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 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.