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.
