Lagrangian concordance is not a partial order 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, $n>1$.

arXiv comment5 pages, 1 figure; updated version, accepted for publication in Pacific Journal of Mathematics

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