Embedded contact homology and its applications

TL;DR

Embedded contact homology (ECH) is a Floer homology for contact 3-manifolds, linking topology and contact geometry, with applications to conjectures and obstructions in symplectic topology.

Contribution

This paper provides an overview of ECH and demonstrates its applications to key problems in contact and symplectic topology.

Findings

ECH is isomorphic to Seiberg-Witten Floer homology.

ECH applications include generalizations of the Weinstein and Arnold conjectures.

ECH provides obstructions to symplectic embeddings in four dimensions.

Abstract

Embedded contact homology (ECH) is a kind of Floer homology for contact three-manifolds. Taubes has shown that ECH is isomorphic to a version of Seiberg-Witten Floer homology (and both are conjecturally isomorphic to a version of Heegaard Floer homology). This isomorphism allows information to be transferred between topology and contact geometry in three dimensions. In this article we first give an overview of the definition of embedded contact homology. We then outline its applications to generalizations of the Weinstein conjecture, the Arnold chord conjecture, and obstructions to symplectic embeddings in four dimensions.