Weak and strong fillability of higher dimensional contact manifolds

TL;DR

This paper extends the concepts of weak and strong symplectic fillability from three-dimensional contact manifolds to higher dimensions, introducing new notions and examples that distinguish fillability from overtwistedness.

Contribution

It defines a higher-dimensional generalization of weak fillings, proves it is weaker than existing notions, and provides the first examples of non-fillable, non-overtwisted contact manifolds in all dimensions.

Findings

Higher-dimensional weak fillability is strictly weaker than strong fillability.

Introduces a higher-dimensional analogue of Giroux torsion.

Constructs examples of non-fillable, non-overtwisted 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.