Positive Legendrian isotopies and Floer Theory

TL;DR

This paper explores positive Legendrian isotopies using Floer homology, revealing new obstructions to positive loops and providing insights into the orderability of contact manifolds through contact homology and symplectic homology.

Contribution

It introduces novel Floer-theoretic obstructions to positive Legendrian loops and connects contact and symplectic homology to orderability of contact manifolds.

Findings

New obstructions to positive Legendrian loops via Floer homology

Identification of orderable contact manifolds using contact and symplectic homology

Examples of contact manifolds with strong orderability properties

Abstract

Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of the Legendrian contact homology of the Legendrian submanifold. As applications, old and new examples of orderable contact manifolds are obtained and discussed. We also show that contact manifolds admitting a filling of a Liouville domain with non-zero symplectic homology is strongly orderable in the sense of Liu.