Topological methods in 3-dimensional contact geometry

TL;DR

This paper introduces Giroux's convex surface theory in contact 3-manifolds, emphasizing explicit examples and applications such as reproofs of key theorems about tight contact structures on S^3.

Contribution

It provides an accessible introduction to convex surface techniques and demonstrates their use in proving fundamental results in contact topology.

Findings

Reproves Bennequin's theorem on tightness of the standard contact structure on S^3

Reproves Eliashberg's uniqueness theorem for tight contact structures on S^3

Highlights the role of finitely many curves in encoding contact structures near surfaces

Abstract

These notes provide an introduction to Giroux's theory of convex surfaces in contact 3-manifolds and its simplest applications. They put a special emphasis on pictures and discussions of explicit examples. The first goal is to explain why all the information about a contact structure in a neighborhood of a generic surface is encoded by finitely many curves on the surface. Then we describe the bifurcations that happen in generic families of surfaces. As applications, we explain how Giroux used this technology to reprove Bennequin's theorem saying that the standard contact structure on S^3 is tight and Eliashberg's theorem saying that all tight contact structures on S^3 are isotopic to the standard one.