Rohlin's invariant and gauge theory II. Mapping tori

Abstract

This is the second in a series of papers studying the relationship between Rohlin’s theorem and gauge theory. We discuss an invariant of a homology $S^1\times S^3$ defined by Furuta and Ohta as an analogue of Casson’s invariant for homology 3–spheres. Our main result is a calculation of the Furuta–Ohta invariant for the mapping torus of a finite-order diffeomorphism of a homology sphere. The answer is the equivariant Casson invariant (Collin–Saveliev 2001) if the action has fixed points, and a version of the Boyer–Nicas (1990) invariant if the action is free. We deduce, for finite-order mapping tori, the conjecture of Furuta and Ohta that their invariant reduces mod 2 to the Rohlin invariant of a manifold carrying a generator of the third homology group. Under some transversality assumptions, we show that the Furuta–Ohta invariant coincides with the Lefschetz number of the action on Floer homology. Comparing our two answers yields an example of a diffeomorphism acting trivially on the representation variety but non-trivially on Floer homology.