ABSTRACT
One of the most interesting problems in low-dimensional topology is the classification of simply-connected closed four-manifolds up to diffeomorphism. This chapter gives a short introduction to Seiberg-Witten invariants. It analyses the local structure of the Seiberg-Witten moduli space. The chapter shows that the moduli space is generically a smooth, oriented manifold. The solution space for the Seiberg-Witten equations is usually infinite-dimensional and one has to mod out by an appropriate infinite-dimensional group, the gauge group, to get a finite-dimensional moduli space. In Donaldson theory one has to analyze the ends of the moduli space and has to use an appropriate compactification in order to define the Donaldson invariants. In Seiberg-Witten theory it for free. Fortunately not all four-manifolds have trivial Seiberg-Witten invariants. Another interesting area is the study of embedded surfaces in four-manifolds. The quickly emerging Seiberg-Witten theory made it possible to push these techniques further and led quickly to the proof of the Thom conjecture.
