Contact 3-Manifolds

Abstract

This chapter is devoted to the recollection of basic facts about contact manifolds. As before, we start with the general case, but very quickly specialize to 3-manifolds. To understand the topology of contact 3-manifolds we consider submanifolds and the contact structures near them. The contact version of Darboux’s theorem says that every point in a contact 3-manifold has a neighborhood which is standard regardless of the contact structure. Then we consider knots which are always tangent or always transverse to the contact planes and examine their classical invariants. It turns out that the contact structures near these types of knots are essentially unique. For an arbitrary surface embedded in a contact 3-manifold we look at the characteristic foliation induced by the contact structure to extract information. It is typical to move a surface by a small isotopy to modify its characteristic foliation to get a generic picture and/or to eliminate certain type of singularities. As it turns out, the characteristic foliation determines the contact structure near the surface. A more complete treatment of the ideas and theorems collected here can be found in e.g. [1, 39, 56, 57].