Seiberg-Witten invariants for 3-manifolds in the case b1 = 0 or 1

Abstract

In this note we give a detailed exposition of the Seiberg-Witten invariants for closed oriented 3-manifolds paying particular attention to the case of b1 = 0 and b1 = 1. These are extracted from the moduli space of solutions to the Seiberg-Witten equations which depend on choices of a Riemannian metric on the underlying manifold as well as certain perturbation terms in the equations. In favourable circumstances this moduli space is finite and naturally oriented and we may form the algebraic sum of the points. Given any two sets of choices of metric and perturbation which are connected by a 1-parameter family, we analyse in detail the singularities which may develop in the interpolating moduli space. This leads then to an understanding of how the algebraic sum changes. In the case b1 = 0 a topological invariant can be extracted with the addition of a suitable counter-term, which we identify (this idea is attributed to Donaldson). In the case b1 = 1 a topological invariant is defined which depends only on cohomological information related to the perturbation term. We prove a 'wall-crossing' formula which tells us how the invariant changes with different choices of this perturbation. Throughout we pay careful attention to genericity statements and the issue of orientations and signs in all the relations. The equivalence of this invariant in the case of an integral homology sphere with the Casson invariant is treated in Lim, 1999 (see also works of Nicolescu, preprint). The equivalence with Reidemeister Torsion in the case b1 > 0 is a result of Meng & Taubes, 1996. Some related material is in Marcolli, 1996, Froyshov, 1996 and in the survey Donaldson, 1996. Taubes, 1990 contains the originating construction in this article in the context of flat SU(2)-connections.