#### Geometry & Topology, Vol. 9 (2005)
Paper no. 47, pages 2079--2127.

## Rohlin's invariant and gauge theory III. Homology $4$--tori

### Daniel Ruberman, Nikolai Saveliev

**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

**AMS subject classification**.
Primary: 57R57.
Secondary: 57R58.

**E-print:** `arXiv:math.GT/0404162`

**DOI:** 10.2140/gt.2005.9.2079

Submitted to GT on 2 August 2005.
Paper accepted 25 October 2005.
Paper published 27 October 2005.

Notes on file formats
Daniel Ruberman, Nikolai Saveliev

Department of Mathematics, MS 050, Brandeis University

Waltham, MA 02454, USA

and

Department of Mathematics, University of Miami

PO Box 249085, Coral Gables, FL 33124, USA

Email: ruberman@brandeis.edu, saveliev@math.miami.edu

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