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.

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Daniel Ruberman, Nikolai Saveliev
Department of Mathematics, MS 050, Brandeis University
Waltham, MA 02454, USA
Department of Mathematics, University of Miami
PO Box 249085, Coral Gables, FL 33124, USA

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