ABSTRACT
In the present chapter, we shall give an introduction to the theory of three-manifolds. The aim of this chapter is to show how different branches of low-dimensional topology can interact and give very strong results. In the first part, we shall describe a sympathetic theory of knots in RP 3. The second part will be devoted to the deep construction of Witten invariants of three-manifolds. We are also going to describe some fundamental constructions of three-dimensional topology, such as the Heegaard decomposition and the Kirby moves. On one hand, they are necessary for constructing the Witten theory; on the other hand, they have their own remarkable interest. The theory of Witten invariants, which is based on the Kauffman bracket on one hand and the Kirby theory on the other, is very deep. In fact, it leads to the theory of invariants of knots in three-manifolds. Many proofs will be sketched or omitted (e.g. the Kirby theorem). For the study of three-manifolds, we recommend Matveev’s book [Matv4].
