Loose Legendrians and the plastikstufe

Abstract

Abstract. We show that the presence of a plastikstufe induces a certain degree of flexibility in contact manifolds of dimension 2n + 1 > 3. More precisely, we prove that every Legendrian knot whose complement contains a "nice" plastikstufe can be destabilized (and, as a consequence, is loose). As an application, it follows in certain situations that two non-isomorphic contact structures become isomorphic after connectsumming with a manifold containing a plastikstufe.

The topology of Stein fillable manifolds in high dimensions I

“…Motivated by their three-dimensional analogues, various notions of fillability and overtwistedness of contact structures on higher-dimensional manifolds have been extensively studied, cf. [34,36]. The class of contact structures satisfying an appropriate h-principle (as overtwisted three-dimensional contact manifolds do), however, has not yet been identified ‡ .…”

On plastikstufe, bordered Legendrian open book and overtwisted contact structures

“…In particular we focus on the following (a bit technical) criterion for overtwistedness. One of the main goals of this paper is to remove the technical conditions of the following theorem from , whose proof relies on the earlier work of . Theorem A contact manifold is overtwisted if it contains a small plastikstufe with spherical core and trivial rotation.…”

“…Let us remark that the requirement of the plastikstufe to be small, with spherical core and trivial rotation as stated in Theorem is purely technical and will not be needed in this paper. So we refer the interested reader to for more details. It should be mentioned that by the work of Murphy, Niederkrüger, Plamenevskaya and Stipsicz , it is known that a Legendrian submanifold is loose, in the sense of Murphy , if its complement contains an embedded small plastikstufe with spherical core and trivial rotation.…”

“…So we refer the interested reader to for more details. It should be mentioned that by the work of Murphy, Niederkrüger, Plamenevskaya and Stipsicz , it is known that a Legendrian submanifold is loose, in the sense of Murphy , if its complement contains an embedded small plastikstufe with spherical core and trivial rotation. Then Theorem follows from the characterization of overtwistedness using loose Legendrian unknot.…”

“…So we refer the interested reader to [26] for more details. It should be mentioned that by the work of Murphy, Niederkrüger, Plamenevskaya and Stipsicz [26], it is known that a Legendrian submanifold is loose, in the sense of Murphy [25], if its complement contains an…”