#### Volume 9, issue 4 (2005)

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

### Daniel Ruberman and Nikolai Saveliev

Geometry & Topology 9 (2005) 2079–2127
 arXiv: math.GT/0404162
##### 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.

##### Keywords
Rohlin invariant, Donaldson invariant, equivariant perturbation, homology torus
Primary: 57R57
Secondary: 57R58