On the monopole Lefschetz number

Let f be a finite order diffeomorphism of a rational homology 3-sphere M making it into an n-fold cyclic branched cover of a knot in an integral homology sphere. We prove a formula for the Lefschetz number of the map induced by f on the reduced monopole homology of M. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg-Witten invariants of 4-manifolds. It has various applications in low-dimensional topology, gauge theory, knot theory, and contact geometry. This is a joint project with Jianfeng Lin and Daniel Ruberman.