  This is a survey of our recent work with Tom Mrowka on Seiberg-Witten gauge
theory and index theory for manifolds with periodic ends. We explain how this
work leads to a new invariant, which is related to the classical Rohlin
invariant of homology 3-spheres and to the Furuta-Ohta invariant originating in
Yang-Mills gauge theory. We give some new calculations of our invariant for
4-dimensional mapping tori.
