Lagrangian concordance is not a partial order in high dimensions

TL;DR

This paper demonstrates that in high-dimensional contact geometry, Lagrangian concordance relations between Legendrian submanifolds are not antisymmetric, thus not forming a partial order, by providing explicit counterexamples.

Contribution

The authors construct explicit examples of Legendrian submanifolds in high dimensions where Lagrangian concordance is bidirectional, showing it does not define a partial order.

Findings

Lagrangian concordances can be bidirectional between non-isotopic Legendrian submanifolds.

This bidirectionality contradicts the antisymmetry property of a partial order.

Lagrangian concordance relations are not partial orders in high dimensions.

Abstract

In this short note we provide the examples of pairs of closed, connected Legendrian non-isotopic Legendrian submanifolds $(\Lambda_{-}, \Lambda_{+})$ of the $(4n+1)$-dimensional contact vector space, $n>1$, such that there exist Lagrangian concordances from $\Lambda_-$ to $\Lambda_+$ and from $\Lambda_+$ to $\Lambda_-$. This contradicts anti-symmetry of the Lagrangian concordance relation, and, in particular, implies that Lagrangian concordances with connected Legendrian ends do not define a partial order in high dimensions. In addition, we explain how to get the same result for the relation given by exact Lagrangian cobordisms with connected Legendrian ends in the $(2n+1)$-dimensional contact vector space, $n>1$.