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 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 S3 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.

Let Y be a closed and oriented 3-manifold. We define different versions of unfolded Seiberg-Witten Floer spectra for Y. These invariants generalize Manolescu's Seiberg-Witten Floer spectrum for rational homology 3-spheres. We also compute some examples when Y is a Seifert space.

We use the construction of unfolded Seiberg-Witten Floer spectra of general 3-manifolds defined in our previous paper to extend the notion of relative Bauer-Furuta invariants to general 4-manifolds with boundary. One of the main purposes of this paper is to give a detailed proof of the gluing theorem for the relative invariants.

No abstract uploaded!

No abstract uploaded!