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
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.