Weak and strong fillability of higher dimensional contact manifolds

Abstract

For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five),while also being obstructed by all known manifestations of "overtwistedness". We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher-dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.

68 pages, 5 figures. v2: Some attributions clarified, and other minor edits. v3: exposition improved using referee's comments. Published by Invent. Math

References in corpus (3)

• The plastikstufe - a generalization of the overtwisted disk to higher dimensions
• Four-dimensional symplectic cobordisms containing three-handles
• Every contact manifold can be given a non-fillable contact structure

Cited by in corpus (35)

• Contact homology and virtual fundamental cycles
• Symplectic homology and the Eilenberg-Steenrod axioms
• The topology of Stein fillable manifolds in high dimensions I
• Maximal contact and symplectic structures
• Loose Legendrians and the plastikstufe
• The topology of Stein fillable manifolds in high dimensions II
• Gromov compactness for holomorphic discs with totally real boundary conditions
• Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology
• Higher symplectic capacities
• Symplectic fillings of asymptotically dynamically convex manifolds I
• Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity
• Symplectic fillings of asymptotically dynamically convex manifolds II--$k$-dilations
• Polyfolds, Cobordisms, and the strong Weinstein conjecture
• Examples of non-trivial contact mapping classes in all dimensions
• On some examples and constructions of contact manifolds
• Pseudorotations of the 2-disc and Reeb flows on the 3-sphere
• $({\mathbb{RP}}^{2n-1},{\xi }_{std})$ is not exactly fillable for $n\ne 2^k$
• On plastikstufe, bordered Legendrian open book and overtwisted contact structures
• Periodic orbits in virtually contact structures
• On properties of Bourgeois contact structures
• Contact surgery and symplectic caps
• On fillings of contact links of quotient singularities
• Higher dimensional contact topology via holomorphic disks
• Contact Hypersurfaces in Uniruled Symplectic Manifolds Always Separate
• Anosov flows and Liouville pairs in dimension three
• Algebraic and Giroux torsion in higher-dimensional contact manifolds
• On the cohomology ring of symplectic fillings
• Planarity in higher-dimensional contact manifolds
• Disjoinable Lagrangian tori and semisimple symplectic cohomology
• Stabilized convex symplectic manifolds are Weinstein
• SFT computations and intersection theory in higher-dimensional contact manifolds
• Examples of non-trivial contact mapping classes for overtwisted contact manifolds in all dimensions
• A remark about weak fillings
• A landscape of contact manifolds via rational SFT
• Connected sums of Brieskorn contact $5$-spheres