Lagrangian concordance is not a partial order in high dimensions

Roman Golovko

Vol. 334 (2025), No. 1, 183–187

DOI: 10.2140/pjm.2025.334.183

Abstract

We provide examples of pairs of closed, connected Legendrian nonisotopic 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 antisymmetry 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, $n>1$.

Keywords

Legendrian submanifold, Lagrangian concordance, partial order

Mathematical Subject Classification

Primary: 53D12

Secondary: 53D42

Milestones

Received: 8 January 2025

Revised: 13 January 2025

Accepted: 16 January 2025

Published: 31 January 2025