Floer homology and Lagrangian concordance

TL;DR

This paper investigates constraints on Lagrangian concordances between Legendrian submanifolds, showing they induce isomorphisms in contact cohomology, and classifies certain concordances in the 3-sphere, revealing new non-invertible examples.

Contribution

It establishes that exact Lagrangian concordances induce isomorphisms in bilinearised Legendrian contact cohomology and classifies all such concordances from the unknot to itself in the 3-sphere.

Findings

Existence of non-invertible exact Lagrangian concordances in all dimensions.

Complete classification of concordances from the Legendrian unknot to itself in the tight contact 3-sphere.

Concordances induce isomorphisms on bilinearised Legendrian contact cohomology.

Abstract

We derive constraints on Lagrangian concordances from Legendrian submanifolds of the standard contact sphere admitting exact Lagrangian fillings. More precisely, we show that such a concordance induces an isomorphism on the level of bilinearised Legendrian contact cohomology. This is used to prove the existence of non-invertible exact Lagrangian concordances in all dimensions. In addition, using a result of Eliashberg-Polterovich, we completely classify exact Lagrangian concordances from the Legendrian unknot to itself in the tight contact-three sphere: every such concordance is the trace of a Legendrian isotopy. We also discuss a high dimensional topological result related to this classification.