Rohlin's invariant and gauge theory III. Homology 4–tori

Abstract

This is the third in our series of papers relating gauge theoretic invariants of certain $4$–manifolds with invariants of $3$–manifolds derived from Rohlin’s theorem. Such relations are well-known in dimension three, starting with Casson’s integral lift of the Rohlin invariant of a homology sphere. We consider two invariants of a spin $4$–manifold that has the integral homology of a $4$–torus. The first is a degree zero Donaldson invariant, counting flat connections on a certain $SO\left(3\right)$–bundle. The second, which depends on the choice of a 1–dimensional cohomology class, is a combination of Rohlin invariants of a $3$–manifold carrying the dual homology class. We prove that these invariants, suitably normalized, agree modulo 2, by showing that they coincide with the quadruple cup product of 1–dimensional cohomology classes.