Introduction to contact topology

Abstract

This master’s thesis provides an introduction to contact topology, with the primary objective of proving Martinet’s Theorem, which asserts that every closed, connected 3-manifold admits a contact structure. The proof heavily relies on the Lickorish-Wallace Theorem, which states that any such 3-manifold can be obtained from $S^{3}$ via a finite sequence of Dehn surgeries. The thesis explores key concepts in contact topology, such as contact structures, Darboux’s Theorem, and Gray stability. A complete proof of the Lickorish-Wallace Theorem is given before focusing on the detailed proof of Martinet’s Theorem, highlighting the ubiquity of contact structures in 3-manifolds.