Non-loose Legendrian spheres with trivial Contact Homology DGA

TL;DR

This paper constructs examples of non-loose Legendrian spheres with trivial Legendrian contact homology DGA, challenging the assumption that trivial DGA implies looseness.

Contribution

It provides explicit non-loose Legendrian spheres that have trivial contact homology DGA, showing trivial DGA does not imply looseness.

Findings

Non-loose Legendrian spheres with trivial DGA exist.

Trivial DGA does not necessarily indicate looseness.

Examples challenge previous assumptions about Legendrian invariants.

Abstract

Loose Legendrian n-submanifolds, for n at least 2, were introduced by Murphy and proved to be flexible in the h-principle sense: any two loose Legendrian submanifolds that are formally Legendrian isotopic are also actually Legendrian isotopic. Legendrian contact homology is a Floer theoretic invariant that associates a differential graded algebra (DGA) to a Legendrian submanifold. The DGA of a loose Legendrian submanifold is trivial.