  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
cross 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.
