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Classification of invariant tight contact structures on the 3-space, -ball and -sphere

Mirko Torresani

TL;DR

The paper extends tight contact topology to an equivariant setting under a $\mathbb{Z}/2\mathbb{Z}$-action conjugate to rotation about the $x$-axis, developing equivariant Gray, Giroux, and Darboux theorems and a parallel convex theory. It introduces equivariant tomography for $S^2\times[-1,1]$, proving that invariant tight structures with fixed boundary foliation form a $\mathbb{Z}$-torsor, and defines a twisting difference via loop-space analysis. Using these tools, the authors classify invariant tight contact structures on $\mathbb{R}^3$, the complement of the standard ball, the standard ball itself, and $S^3$, obtaining trivial sets in some cases and $\mathbb{Z}$-torsor structures in others. The work connects to strongly invertible Legendrian knot classification through the introduced twisting torsor and equivariant tomographic framework. Overall, the method combines equivariant transversality, Morse–Smale perturbations, and loop-space techniques to achieve a precise torsor-based classification.

Abstract

We prove some classification results for tight contact structure in the 3-space, -ball and -sphere that are invariant with respect to some arbitrary involution, that is conjugated to the standard rotation around the x-axis. Unlike the classical scenario, a new integral torsion appears, dictating a splitting between equivalence classes. These tools could be useful fur future classification results regarding strongly invertible Legendrian knots.

Classification of invariant tight contact structures on the 3-space, -ball and -sphere

TL;DR

The paper extends tight contact topology to an equivariant setting under a -action conjugate to rotation about the -axis, developing equivariant Gray, Giroux, and Darboux theorems and a parallel convex theory. It introduces equivariant tomography for , proving that invariant tight structures with fixed boundary foliation form a -torsor, and defines a twisting difference via loop-space analysis. Using these tools, the authors classify invariant tight contact structures on , the complement of the standard ball, the standard ball itself, and , obtaining trivial sets in some cases and -torsor structures in others. The work connects to strongly invertible Legendrian knot classification through the introduced twisting torsor and equivariant tomographic framework. Overall, the method combines equivariant transversality, Morse–Smale perturbations, and loop-space techniques to achieve a precise torsor-based classification.

Abstract

We prove some classification results for tight contact structure in the 3-space, -ball and -sphere that are invariant with respect to some arbitrary involution, that is conjugated to the standard rotation around the x-axis. Unlike the classical scenario, a new integral torsion appears, dictating a splitting between equivalence classes. These tools could be useful fur future classification results regarding strongly invertible Legendrian knots.
Paper Structure (9 sections, 47 theorems, 91 equations, 1 figure)

This paper contains 9 sections, 47 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Equivariant Giroux and Darboux' Theorems
  3. Equivariant Convex Theory
  4. Equivariant Tomography Theory
  5. Reconstruction and Uniqueness Lemmas
  6. Defining the twisting torsor
  7. Proof of Theorem thm:mainthmtomographyStwo
  8. Proof of Theorems thmmainthmRthree and thmmainthmDthreecomplementary
  9. Proof of Theorems thmmainthmDthree and thmmainthmSthree

Key Result

Theorem 1

The set of tight contact structures on $\mathbb{R}^3$ up to equivariant isotopy is trivial.

Figures (1)

  • Figure 1: The angle between the oriented planes $T_pS$ and $\xi_p$

Theorems & Definitions (98)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition
  • Definition
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 88 more