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