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Strong closing property of contact forms and action selecting functors

Kei Irie

Abstract

We introduce a notion of strong closing property of contact forms, inspired by the $C^\infty$ closing lemma for Reeb flows in dimension three. We then prove a sufficient criterion for strong closing property, which is formulated by considering a monoidal functor from a category of manifolds with contact forms to a category of filtered vector spaces. As a potential application of this criterion, we propose a conjecture which says that a standard contact form on the boundary of any symplectic ellipsoid satisfies strong closing property.

Strong closing property of contact forms and action selecting functors

Abstract

We introduce a notion of strong closing property of contact forms, inspired by the closing lemma for Reeb flows in dimension three. We then prove a sufficient criterion for strong closing property, which is formulated by considering a monoidal functor from a category of manifolds with contact forms to a category of filtered vector spaces. As a potential application of this criterion, we propose a conjecture which says that a standard contact form on the boundary of any symplectic ellipsoid satisfies strong closing property.
Paper Structure (19 sections, 17 theorems, 53 equations)

This paper contains 19 sections, 17 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Setup and notations
  3. Strong closing property
  4. Results in low dimensions
  5. Contents of this paper
  6. Contact action selectors
  7. Action selecting functors
  8. Category $\text{\rm Cont}_{2n-1}$
  9. Category $\text{\rm FVect}^{{\mathbb Z}/2}_K$
  10. Action selecting functors
  11. Contact action selectors defined from action selecting functors
  12. Category $\text{\rm Cont}^0_{2n-1}$
  13. Examples of action selecting functors
  14. A sufficient criterion for strong closing property
  15. Standard contact sphere
  16. ...and 4 more sections

Key Result

Theorem 1.5

Any contact form on any closed three-manifold satisfies strong closing property.

Theorems & Definitions (50)

  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: Irie_dense
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • ...and 40 more