Legendrian non-isotopic unit conormal bundles in high dimensions

TL;DR

This paper demonstrates that certain high-dimensional Legendrian submanifolds, specifically unit conormal bundles of submanifolds in Euclidean space, can be distinguished by new non-classical invariants despite classical invariants failing.

Contribution

It introduces the use of strip Legendrian contact homology and a coproduct to distinguish Legendrian submanifolds that are topologically similar, expanding the understanding of Legendrian isotopy in high dimensions.

Findings

Examples of non-isotopic Legendrian submanifolds distinguished by coproduct

Development of topological description of invariants for high codimension

Application of string topology to compute invariants

Abstract

For any compact connected submanifold $K$ of $\mathbb{R}^n$, let $\Lambda_K$ denote its unit conormal bundle, which is a Legendrian submanifold of the unit cotangent bundle of $\mathbb{R}^n$. In this paper, we give examples of pairs $(K_0,K_1)$ of compact connected submanifolds of $\mathbb{R}^n$ such that $\Lambda_{K_0}$ is not Legendrian isotopic to $\Lambda_{K_1}$, although they cannot be distinguished by classical invariants. Here, $K_1$ is the image of an embedding $\iota_f \colon K_0 \to \mathbb{R}^n$ which is regular homotopic to the inclusion map of $K_0$ and the codimension in $\mathbb{R}^n$ is greater than or equal to $4$. As non-classical invariants, we define the strip Legendrian contact homology and a coproduct on it under certain conditions on Legendrian submanifolds. Then, we give a purely topological description of these invariants for $\Lambda_K$ when the codimension of $K$ is greater than or equal to $4$. The main examples $\Lambda_{K_0}$ and $\Lambda_{K_1}$ are distinguished by the coproduct, which is computed by using an idea of string topology.