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


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.

Rohlin invariant, Donaldson invariant, equivariant perturbation, homology torus
Mathematical Subject Classification 2000
Primary: 57R57
Secondary: 57R58
Forward citations
Received: 2 August 2005
Accepted: 25 October 2005
Published: 27 October 2005
Proposed: Ronald Stern
Seconded: Ronald Fintushel, Simon Donaldson
Daniel Ruberman
Department of Mathematics
MS 050
Brandeis University
MA 02454
Nikolai Saveliev
Department of Mathematics
University of Miami
PO Box 249085
Coral Gables
Florida 33124