Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tight and overtwisted contact structures

John B. Etnyre

TL;DR

This survey traces the tight versus overtwisted dichotomy in 3D contact geometry, beginning with the historical lineage from Martinet, Lutz, and Bennequin and culminating in Eliashberg’s definitive OT classification and tightness criteria. It highlights the Eliashberg–Thurston construction that deforms foliations into tight contact structures, surveys the current landscape of tight versus OT manifolds (including finiteness and non-fillable examples), and surveys powerful applications such as contact surgery, near-symplectic structures, and Engel structures. The article also surveys higher-dimensional developments, notably the Borman–Eliashberg–Murphy generalization of OT theory and its strong h-principle flavor, signaling a broad, flexible framework for contact topology beyond dimension three. Together, these threads show OT and tight structures as central organizing tools driving modern low- and high-dimensional topology, with far-reaching structural and constructive consequences across geometry and topology.

Abstract

The tight versus overtwisted dichotomy has been an essential organizing principle and driving force in 3-dimensional contact geometry since its inception around 1990. In this article, we will discuss the genesis of this dichotomy in Eliashberg's seminal work and his influential contributions to the theory.

Tight and overtwisted contact structures

TL;DR

This survey traces the tight versus overtwisted dichotomy in 3D contact geometry, beginning with the historical lineage from Martinet, Lutz, and Bennequin and culminating in Eliashberg’s definitive OT classification and tightness criteria. It highlights the Eliashberg–Thurston construction that deforms foliations into tight contact structures, surveys the current landscape of tight versus OT manifolds (including finiteness and non-fillable examples), and surveys powerful applications such as contact surgery, near-symplectic structures, and Engel structures. The article also surveys higher-dimensional developments, notably the Borman–Eliashberg–Murphy generalization of OT theory and its strong h-principle flavor, signaling a broad, flexible framework for contact topology beyond dimension three. Together, these threads show OT and tight structures as central organizing tools driving modern low- and high-dimensional topology, with far-reaching structural and constructive consequences across geometry and topology.

Abstract

The tight versus overtwisted dichotomy has been an essential organizing principle and driving force in 3-dimensional contact geometry since its inception around 1990. In this article, we will discuss the genesis of this dichotomy in Eliashberg's seminal work and his influential contributions to the theory.

Paper Structure

This paper contains 12 sections, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Precursors and the birth of modern contact topology
  3. The classification of overtwisted contact $3$-manifolds
  4. Legacy and applications of overtwisted contact structures
  5. Results about overtwisted contact structures
  6. Applications of overtwisted contact structures
  7. Legacy and applications of tight contact structures
  8. The Eliashberg-Thurston theorem
  9. The current state of tight contact structures
  10. Applications of tight contact structures
  11. Higher dimensional overtwisted contact structures
  12. Final thoughts

Figures (2)

  • Figure 1: Yasha at a conference in the late 1970s when several of the ideas leading to the tight versus overtwisted dichotomy were being developed.
  • Figure 4: On the left, each dot represents a non-loose Legendrian unknot and the arrows indicate what happens after stabilization. On the left we see the same for Legendrian unknots in the standard tight contact structure on $S^3$.