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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