The Giroux Correspondence in dimension 3

TL;DR

This paper establishes the Giroux Correspondence in three dimensions by utilizing Heegaard splittings of contact manifolds, extending previous tight case results to all contact 3-manifolds for broader accessibility.

Contribution

It extends the Heegaard splitting approach to prove Giroux Correspondence for all contact 3-manifolds, not just tight ones, making the proof more accessible.

Findings

Proves Giroux Correspondence for all contact 3-manifolds.

Introduces classification moves for bypass decompositions.

Provides an accessible proof for low-dimensional topology.

Abstract

This paper proves the Giroux Correspondence in dimension three using Heegaard splittings of contact manifolds. In two of the authors earlier paper they proved the Giroux Correspondence for tight contact 3-manifolds via convex Heegaard surfaces, and simultaneously, Honda, Breen and Huang gave an alldimensions proof of the Giroux Correspondence by generalising convex surface theory to higher dimensions. This paper extends the Heegaard splitting approach to arbitrary (not necessarily tight) contact 3-manifolds in order to provide a proof accessible to a low-dimensional audience. The proof assumes classification moves relating bypass decompositions for isotopic contact structures on cobordisms that are topological products; in the Appendix, we prove this result in the 3- dimensional setting.