Vanishing of the contact homology of overtwisted contact 3--manifolds

TL;DR

This paper proves that overtwisted contact structures on closed 3-manifolds have trivial contact homology, confirming a theorem by Eliashberg with two different proofs.

Contribution

It provides a new proof and an alternative outline for the vanishing of contact homology in overtwisted contact 3-manifolds, confirming a key theorem.

Findings

Contact homology vanishes for overtwisted structures

Two independent proofs are provided

Confirms Eliashberg's theorem on contact homology

Abstract

We give a proof of, for the case of contact structures defined by global contact 1-forms, a Theorem stated by Eliashberg that for any overtwisted contact structure on a closed 3-manifold, its contact homology is 0. A different proof is also outlined in the appendix by Yakov Eliashberg.