On positive loops of loose Legendrian embeddings

TL;DR

This paper demonstrates the existence of contractible positive loops of Legendrian embeddings for loose Legendrian submanifolds using the h-principle, and explores implications for contact topology orderability.

Contribution

It establishes the existence of positive loops for loose Legendrians and introduces a new partial order on contactomorphism groups, showing overtwisted manifolds are weakly non-orderable.

Findings

Existence of contractible positive loops for loose Legendrians.

Definition of a new partial order on contactomorphism groups.

Overtwisted contact manifolds are weakly non-orderable.

Abstract

In this paper, via h-principle we prove that there exist contractible positive loops of Legendrian embeddings based at any loose Legendrian submanifold. As an application, we define a new partial order on $\widetilde{Cont}_0(M,\xi)$ and prove that overtwisted contact manifolds are weakly non-orderable.