Three-manifold topology and the Donaldson-Witten partition function Original Research Article

We consider Donaldson-Witten theory on four-manifolds of the form X = Y × S1 where Y is a compact three-manifold. We show that there are interesting relations between the four-dimensional Donaldson invariants of X and certain topological invariants of Y. In particular, we reinterpret a result of Meng-Taubes relating the Seiberg-Witten invariants to Reidemeister-Milnor torsion. If b1(Y) > 1 we show that the partition function reduces to the Casson-Walker-Lescop invariant of Y, as expected on formal grounds. In the case b1(Y) = 1 there is a correction. Consequently, in the case b1(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.