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

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

2004-04-07

arxiv.org/abs/math/0404162v3

Geom. Topol. 9(2005) 2079-2127

Mathematics

Geometric Topology

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper47.abs.html

Scientific paper

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.

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