Legendrian non-isotopic unit conormal bundles in high dimensions

Abstract

For any compact connected submanifold $K$ of ${\mathbb{R}}^n$, let $\hat{I}{›}_K$ 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 $\hat{I}{›}_{K_0}$ is not Legendrian isotopic to $\hat{I}{›}_{K_1}$, although they cannot be distinguished by classical invariants. Here, $K_1$ is the image of an embedding ${\hat{I}}_f^1:K_0\to {\mathbb{R}}^n$ 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 $\hat{I}{›}_K$ when the codimension of $K$ is greater than or equal to $4$. The main examples $\hat{I}{›}_{K_0}$ and $\hat{I}{›}_{K_1}$ are distinguished by the coproduct, which is computed by using an idea of string topology.

66 pages, 13 figures, revised following referee's comments