Loose Legendrians and the plastikstufe

TL;DR

This paper demonstrates that the presence of a plastikstufe in high-dimensional contact manifolds introduces flexibility, allowing Legendrian knots to be destabilized and making certain contact structures become equivalent after connect-summing.

Contribution

It establishes a link between plastikstufe presence and flexibility in high-dimensional contact topology, showing how it affects Legendrian knots and contact structure isomorphisms.

Findings

Legendrian knots with a 'nice' plastikstufe can be destabilized

Presence of a plastikstufe induces flexibility in contact manifolds

Certain contact structures become isomorphic after connect-summing with a plastikstufe manifold

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 connect-summing with a manifold containing a plastikstufe.