On the monopole Lefschetz number of finite-order diffeomorphisms

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 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.