Three-manifold topology and the Donaldson-Witten partition function

We consider Donaldson-Witten theory on four-manifolds of the form X = Y x S -1 where Y is a compact three-manifold. We show that there are interesting relations between the four-dimensional Donaldson invariants of X and certai n topological invariants of Y. In particular, we reinterpret a result of Me ng-Taubes relating the Seiberg-Witten invariants to Reidemeister-Milnor tor sion, If b(1)(Y) > 1 we show that the partition function reduces to the Cas son-Walker-Lescop invariant of Y, as expected on formal grounds. In the cas e b(1)(Y) = 1 there is a correction. Consequently, in the case b(1)(Y) = 1, we observe an interesting subtlety in the standard expectations of Kaluza- Klein theory when applied to supersymmetric gauge theory compactified on a circle of small radius. (C) 1999 Published by Elsevier Science B.V.