Loose Legendrians and the plastikstufe

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 nonisomorphic contact structures become isomorphic after connect-summing with a manifold containing a plastikstufe.