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Tight contact structures without symplectic fillings are everywhere

Jonathan Bowden, Fabio Gironella, Agustin Moreno, Zhengyi Zhou

Abstract

We show that for all $n \geq 3$, any $(2n+1)$-dimensional manifold that admits a tight contact structure, also admits a tight but non-fillable contact structure, in the same almost contact class. For $n=2$, we obtain the same result, provided that the first Chern class vanishes. We further construct Liouville but not Weinstein fillable contact structures on any Weinstein fillable contact manifold of dimension at least $7$ with torsion first Chern class.

Tight contact structures without symplectic fillings are everywhere

Abstract

We show that for all , any -dimensional manifold that admits a tight contact structure, also admits a tight but non-fillable contact structure, in the same almost contact class. For , we obtain the same result, provided that the first Chern class vanishes. We further construct Liouville but not Weinstein fillable contact structures on any Weinstein fillable contact manifold of dimension at least with torsion first Chern class.
Paper Structure (17 sections, 23 theorems, 27 equations, 1 figure)

This paper contains 17 sections, 23 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bourgeois contact manifolds
  3. Reeb dynamics of Bourgeois contact structures
  4. ADC Bourgeois contact manifolds
  5. $\mathbb{S}^1-$invariant contact structures and convex decompositions
  6. Capping Giroux domains
  7. Allowing surgeries on a Giroux domain
  8. An algebraic perspective on tightness
  9. Obstructions to strong fillings
  10. Homological restrictions on symplectic fillings
  11. Symplectic Cohomology
  12. Basics of symplectic cohomology
  13. Neck-stretching
  14. Non-fillable contact structures
  15. Tight non-fillable structures on spheres
  16. ...and 2 more sections

Key Result

Theorem 1

If $(M^{2n+1},\xi)$ with $n \ge 3$ is tight, then $M$ admits a tight, non-fillable contact structure in the same almost contact class. If $n =2$, the same holds, with the extra assumption that the first Chern class of $\xi$ is zero.

Figures (1)

  • Figure 1: The blow-down handle $V_-\times D^2$ in green, attached on top of $\{1\}\times V_-\times \mathbb{S}^1 \subset \{1\}\times M$ seen as the boundary of the trivial cobordism $[0,1]\times M$. The purple region depicts the handle attachment that gives $M'$ from $M$, attached on the region $V_+\times \mathbb{S}^1\subset M$. In red the contact manifold $M'_+$ resulting from both blow-down and contact surgeries.

Theorems & Definitions (51)

  • Theorem 1
  • Theorem 2
  • Remark
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 2.1: Bourgeois Bo
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 41 more