Geometry & Topology 9 (2005) 2079–2127

DOI: 10.2140/gt.2005.9.2079

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(3)–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

Mathematical Subject Classification

Primary: 57R57

Secondary: 57R58

References

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Publication

Received: 2 August 2005
Accepted: 25 October 2005
Published: 27 October 2005
Proposed: Ronald Stern
Seconded: Ronald Fintushel, Simon Donaldson

Authors