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Homological invariants of codimension 2 contact submanifolds

Laurent Côté, François-Simon Fauteux-Chapleau

Abstract

Codimension 2 contact submanifolds are the natural generalization of transverse knots to contact manifolds of arbitrary dimension. In this paper, we construct new invariants of codimension 2 contact submanifolds. Our main invariant can be viewed as a deformation of the contact homology algebra of the ambient manifold. We describe various applications of these invariants to contact topology. In particular, we exhibit examples of codimension 2 contact embeddings into overtwisted and tight contact manifolds which are formally isotopic but fail to be isotopic through contact embeddings. We also give new obstructions to certain relative symplectic and Lagrangian cobordisms.

Homological invariants of codimension 2 contact submanifolds

Abstract

Codimension 2 contact submanifolds are the natural generalization of transverse knots to contact manifolds of arbitrary dimension. In this paper, we construct new invariants of codimension 2 contact submanifolds. Our main invariant can be viewed as a deformation of the contact homology algebra of the ambient manifold. We describe various applications of these invariants to contact topology. In particular, we exhibit examples of codimension 2 contact embeddings into overtwisted and tight contact manifolds which are formally isotopic but fail to be isotopic through contact embeddings. We also give new obstructions to certain relative symplectic and Lagrangian cobordisms.

Paper Structure

This paper contains 56 sections, 96 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Energy and positivity of intersection
  4. Legendrian invariants and the surgery formula
  5. Applications to contact and Legendrian embeddings
  6. Applications to symplectic and Lagrangian cobordisms
  7. Context and related invariants
  8. Notation and conventions
  9. Acknowledgements
  10. Geometric preliminaries
  11. Symplectic cobordisms
  12. Homotopy classes of asymptotically cylindrical maps
  13. Reeb dynamics near a codimension $2$ contact submanifold
  14. The Conley-Zehnder index
  15. Normal dynamics and adapted contact forms
  16. ...and 41 more sections

Key Result

Theorem 1.7

Let $i, j$ and $(Y, \xi)$ be constructed according to construction:ot-infinite-topology, where $(Y, \xi)$ is an overtwisted contact manifold and $i$ and $j$ are (formally isotopic) contact embeddings. Then $i$ and $j$ are not isotopic through contact embeddings. In fact, $i$ is not isotopic to any r

Theorems & Definitions (306)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Example 1.5: cf. \ref{['definition:girouxform']} and girouxicm
  • Example 1.6: see \ref{['definition:contact-pushoff']} and Def. 3.1 in casals-etnyre
  • Theorem 1.7: see \ref{['theorem:distinguish-infinite-ot']}
  • Theorem 1.9
  • Theorem 1.10
  • Remark 1.11
  • ...and 296 more