Weak and strong fillability of higher dimensional contact manifolds

Abstract

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.

Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity

“…Note that the Reeb orbits that we find in this way are contractible, though there are contact manifolds which are hypertight, meaning that no Reeb vector field has a contractible Reeb orbit. This phenomenon is related to symplectic fillability, see [26] and the references therein. This stronger conclusion should be compared with Corollary 3 which also asserts the existence of a contractible orbit of a Hamiltonian system.…”

On plastikstufe, bordered Legendrian open book and overtwisted contact structures

“…Observe that the singular foliation is nothing but the open book decomposition with page and identity monodromy such that is identified with the binding. Inspired by this observation, we introduce the notion of bordered Legendrian open book, or bLob in short, as defined in . Here we give the definitions in full generality but our results are applied only to bLobs in dimension 5.…”

“…Later, the notion of plastikstufe is generalized by Massot, Niederkrüger and Wendl in to the so‐called bordered Legendrian open books , or bLobs in short, which is, roughly speaking, a compact submanifold of with non‐empty Legendrian boundary and comes with an open book decomposition whose pages are also Legendrian. In particular, the plastikstufe may be regarded as a bLob such that the page is diffeomorphic to the product of the binding and a closed interval.…”

“…It is shown in [, Theorem 4.4] that the presence of an embedded bLob obstructs weak (semipositive) symplectic fillings given a technical cohomological condition on the symplectic form. Combining with Theorem and the overtwisted ‐principle of course, one can easily remove the cohomological condition, at least in dimension 5.…”

Loose Legendrians and the plastikstufe

“…On the one hand, there is a variety of special submanifolds, plastikstufes and bLobs [Ni1,MNW], whose presence in a contact manifold is known to obstruct fillability. On the other hand, certain related objects possess flexibility properties desired for "overtwisted" contact structures.…”