Differential Topology

摘要

The purpose of this chapter is to give a survey of some basic tools needed for studying low dimensional geometric topology. The first section is devoted to an introduction to differential topology. We recall the regular value theorem. Transversality theorem and Whitney embedding theorem are stated in the large context of manifolds with boundary. We discuss orientation, tubular neighborhoods and collars. The end of this section is devoted to the isotopy relation which will play a central part in what follows. In the second section, we apply differential topology to the study of knots and links. We define the diagram of a link, state Reidemeister theorem and give some classical knot invariants. The third section is mainly devoted to Morse theory. We conclude with Heegaard splitting of 3-manifolds and handle decomposition. The next chapter will consider similar notions in the combinatorial context. Most proofs can be found in the classic literature given in the bibliography.