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

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Cited by in corpus (36)

• Contact homology and virtual fundamental cycles
• Symplectic homology and the Eilenberg-Steenrod axioms
• The topology of Stein fillable manifolds in high dimensions I
• A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds
• 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
• Polyfolds, Cobordisms, and the strong Weinstein conjecture
• Symplectic fillings of asymptotically dynamically convex manifolds II--$k$-dilations
• On some examples and constructions of contact manifolds
• Examples of non-trivial contact mapping classes in all dimensions
• $({\mathbb{RP}}^{2n-1},{\xi }_{std})$ is not exactly fillable for $n\ne 2^k$
• On plastikstufe, bordered Legendrian open book and overtwisted contact structures
• Pseudorotations of the 2-disc and Reeb flows on the 3-sphere
• On properties of Bourgeois contact structures
• Periodic orbits in virtually contact structures
• Contact surgery and symplectic caps
• Higher dimensional contact topology via holomorphic disks
• On fillings of contact links of quotient singularities
• Planarity in higher-dimensional contact manifolds
• On the cohomology ring of symplectic fillings
• Anosov flows and Liouville pairs in dimension three
• Algebraic and Giroux torsion in higher-dimensional contact manifolds
• Contact Hypersurfaces in Uniruled Symplectic Manifolds Always Separate
• 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 landscape of contact manifolds via rational SFT
• A remark about weak fillings
• Connected sums of Brieskorn contact $5$-spheres