Lagrangian concordance is not a partial order in high dimensions

TL;DR

The paper shows that in high dimensions ($n>1$), Lagrangian concordances with connected Legendrian ends do not define a partial order on closed, connected Legendrian submanifolds of $\mathbb{R}^{4n+1}_{st}$ by constructing pairs of non-isotopic Legendrian submanifolds that admit concordances in both directions. The argument combines the classification of loose Legendrians, Thurston–Bennequin and rotation invariants, Wu's smooth isotopy result, and the $h$-principle for exact Lagrangian cobordisms with loose ends (via Eliashberg–Murphy) to realize two-way concordances. A parallel construction using non-cylinder exact Lagrangian endocobordisms and front spinning extends the result to exact Lagrangian cobordisms in $\mathbb{R}^{2n+1}_{st}$ for all $n>1$, reinforcing that anti-symmetry fails and the relation is not a partial order. This has implications for the structure of Legendrian cobordism theories and Symplectic Field Theory in higher dimensions.

Abstract

In this short note we provide the examples of pairs of closed, connected Legendrian non-isotopic Legendrian submanifolds $(Λ_{-}, Λ_{+})$ of the $(4n+1)$-dimensional contact vector space, $n>1$, such that there exist Lagrangian concordances from $Λ_-$ to $Λ_+$ and from $Λ_+$ to $Λ_-$. 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, $n>1$.

Figures (1)

• Figure 1: The pair of exact Lagrangian cobordisms $L_{T^2}^{S^2}$ (left) and $L_{S^2}^{T^2}$ (right) from DimitroglouRizellGolovko14.

Theorems & Definitions (6)

• Theorem 1
• Corollary 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6