Lagrangian concordance is not a partial order in high dimensions

Roman Golovko

REVIEW 2 major objections 4 minor 18 references

Lagrangian concordance is not a partial order in high dimensions

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that Lagrangian concordance is not a partial order in high dimensions: non-isotopic Legendrian submanifolds can be Lagrangian concordant in both directions.

desk verdict A plausible and important high-dimensional counterexample to the partial-order question, but the key h-principle step is asserted rather than proved. read the letter →

arxiv 2411.12114 v2 pith:OFPQQCWP submitted 2024-11-18 math.SG

Roman Golovko This is my paper · ORCID

classification math.SG MSC 53D1253D42

keywords LagrangianconcordanceLegendriansubmanifoldpartialorderlooseh-principleexactcobordismfrontspinninganti-symmetry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to establish that, in high dimensions, the relation "there is a Lagrangian concordance from one closed Legendrian submanifold to another" is not a partial order. The author constructs, for each $n>1$, a pair of closed, connected Legendrian submanifolds $\Lambda_-$ and $\Lambda_+$ of the standard contact vector space $\mathbb{R}^{4n+1}_{st}$ that are not Legendrian isotopic, yet admit Lagrangian concordances in both directions. Mutual comparability without isotopy violates antisymmetry, so the concordance relation cannot be a partial order in those dimensions. The same conclusion is drawn for exact Lagrangian cobordisms with connected Legendrian ends in $\mathbb{R}^{2n+1}_{st}$, using examples whose ends are not even diffeomorphic. The result matters because Lagrangian concordance has been a candidate geometric notion of ordering Legendrian submanifolds by complexity.

What carries the argument

The machinery has three parts. First, a classification of loose Legendrian embeddings says that in $\mathbb{R}^{4n+1}$ with $n>1$, for a fixed manifold and rotation class there are exactly two non-isotopic loose Legendrian embeddings up to Legendrian isotopy; this produces the two ends. Second, the Thurston--Bennequin formula shows the two ends share this classical invariant, so they cannot be distinguished by it. Third, the h-principle for exact Lagrangian embeddings with loose concave ends, extended from caps to cobordisms with possibly non-trivial convex ends, converts the smooth concordances supplied by the smooth embedding theorem into exact Lagrangian concordances. Loose ends are what make the h-principle applicable: the formal data can be genuinely realized because the concave ends are loose.

What would settle it

Construct a smooth concordance with loose Legendrian ends that satisfies the formal conditions of the flexibility theorem but provably cannot be deformed to an exact Lagrangian concordance; such an example would invalidate the adaptation in Remark 5 and collapse the proof of Theorem 1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: for every $n>1$, there exists a pair of closed, connected Legendrian submanifolds $\Lambda_-$ and $\Lambda_+$ of the standard contact vector space $\mathbb{R}^{4n+1}_{st}$ that are not Legendrian isotopic, but for which there are Lagrangian concordances $L_\pm$ from $\Lambda_-$ to $\Lambda_+$ and $L_\mp$ from $\Lambda_+$ to $\Lambda_-$. Since antisymmetry would force the two ends to be isotopic, the existence of such a pair implies that the Lagrangian concordance relation is not a partial order on closed, connected Legendrian submanifolds (Corollary 2). The construction takes $\Lambda$ to be any closed, stably parallelizable, simply connected manifold, for instance $S^{2n}$, chooses two loose Legendrian embeddings of $\Lambda$ with the same rotation class that are not Legendrian isotopic, and then uses a classical smooth embedding theorem to obtain smooth concordances in both directions. The h-principle for exact Lagrangian embeddings with loose concave ends is invoked to upgrade these smooth concordances to genuine Lagrangian concordances. The final section repeats the same conclusion for exact Lagrangian cobordisms with connected ends, using a pair of cobordisms between a Legendrian $S^2$ and a Legendrian $T^2$, extended to higher dimensions by front spinning; here the ends are not even diffeomorphic.

Load-bearing premise

The proof relies on an unproven adaptation of a flexibility theorem: the result for exact Lagrangian caps with loose concave ends is assumed to hold for cobordisms with a possibly non-trivial convex end, on the grounds that the relevant homotopies and isotopies are compactly supported.

Editorial extensions

If this is right

Corollary 2 follows directly: Lagrangian concordances with connected Legendrian ends do not define a partial order on closed, connected Legendrian submanifolds of $\mathbb{R}^{4n+1}_{st}$ for $n>1$.
In the exact Lagrangian cobordism setting, anti-symmetry fails in $\mathbb{R}^{2n+1}_{st}$ for all $n>1$; the constructed cobordisms connect Legendrian ends with different diffeomorphism types, such as $S^2$ and $T^2$ after spinning.
The two ends $\Lambda_-$ and $\Lambda_+$ have the same Thurston--Bennequin number and rotation class, and both have acyclic Legendrian contact homology, so those invariants cannot prevent two-way Lagrangian concordance.
The non-partial-order phenomenon occurs even among closed, connected, loose Legendrian submanifolds, not merely in non-compact or disconnected settings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension, not stated in the paper, is that among loose Legendrians in even dimensions the Lagrangian concordance relation may be as flexible as smooth concordance: any smooth concordance with loose ends might be upgradeable to a Lagrangian one.
The paper's method does not directly cover odd-dimensional contact vector spaces, since the classification it relies on gives a unique loose embedding per formal class there; whether two-way Lagrangian concordances exist in those dimensions remains open.
For exact Lagrangian cobordisms, the two-way examples connect non-isotopic and even non-diffeomorphic ends, suggesting that any partial-order structure in that setting would need to compare different topologies rather than merely Legendrian isotopy classes.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is the first high-dimensional example of Lagrangian concordances violating antisymmetry, and the argument is probably correct — but the load-bearing step is asserted, not proved. I would send it to a serious referee, and I would expect that referee to demand a proof of Remark 5 before accepting.

What's new and good: [8] proved non-antisymmetry for Legendrian knots in R^3; this paper extends the phenomenon to all contact dimensions R^{4n+1} with n>1, using closed connected loose Legendrians such as S^{2n}. The strategy is concise and transparent: Murphy's classification produces two loose, non-isotopic Legendrians with identical rotation class and Thurston-Bennequin number; Wu's embedding theorem gives smooth concordances in both directions; and the Eliashberg-Murphy h-principle is supposed to upgrade those to exact Lagrangian concordances. Section 3 handles exact Lagrangian cobordisms with a separate, simpler topological argument via front spinning; that part reads correctly.

The soft spot is Remark 5. Eliashberg-Murphy's Theorem 2.2 is stated for exact Lagrangian caps with a loose concave end. The paper applies it to a cobordism with a loose concave end and a possibly non-trivial convex end, and justifies this by saying the homotopies and isotopies in [11, Theorems 2.2 and 2.3] are compactly supported. That is not a proof. The relative h-principle with two boundary components may well hold, but one needs to show the positive end is fixed as an honest Legendrian embedding and that the formal data at both ends is compatible. Since the upgrade from smooth to Lagrangian concordances is the whole content of Theorem 1, this gap is load-bearing. I do not think the theorem is false; I would bet it is true. But as written, the paper is incomplete.

Other observations: the use of Murphy's Proposition A.4(c) is terse, though Remark 6 helps. The citation pattern is reasonable; the author's own prior work is used for exactly the nuggets it proves. The exposition is clear and the note is short, which is a virtue.

Who this is for: symplectic topologists working on Lagrangian cobordisms and Legendrian classification. The statement is significant enough that the gap is worth trying to close. My recommendation: send it to peer review, and make the referee check Remark 5 carefully. If the adaptation is correct, the paper is publishable essentially as is; if not, it should be rewritten as a conditional result or a conjecture.

Referee Report

2 major / 4 minor

Summary. The paper claims that Lagrangian concordance with connected Legendrian ends is not a partial order on closed, connected Legendrian submanifolds of R^{4n+1}_{st} for n > 1, and that the same holds for the relation given by exact Lagrangian cobordisms in R^{2n+1}_{st}. The proof constructs, for a fixed stably parallelizable simply connected manifold such as S^{2n}, two loose Legendrian embeddings Λ_- and Λ_+ that are not Legendrian isotopic, uses Wu's theorem to obtain smooth concordances between them in both directions, and then invokes an h-principle of Eliashberg–Murphy to upgrade these smooth concordances to exact Lagrangian concordances. The paper also sketches a construction of non-antisymmetric exact Lagrangian cobordisms using previously known examples and the front/spherical spinning construction.

Significance. If the h-principle step is fully justified, the result is a concise and appealing resolution of a natural open question in high-dimensional contact topology, and the auxiliary exact-cobordism statement provides additional context. The proof strategy is coherent and builds on major recent developments (Murphy's classification of loose Legendrians and Eliashberg–Murphy h-principles). However, the central upgrade from smooth to Lagrangian concordances rests on an unproved extension of the cap theorem, so the current manuscript does not yet establish the main theorem with complete rigor.

major comments (2)

[§2, Remark 5] The h-principle extension asserted in Remark 5 is the load-bearing step of the proof: it is the only mechanism that turns the smooth concordances LC∞± and LC∞∓ from Wu's theorem into Lagrangian concordances. [11, Theorem 2.2] is stated for exact Lagrangian caps with a loose concave end and no other boundary, whereas the present proof needs the analogous statement for a cobordism with two boundary components, with the convex end fixed as a genuine Legendrian embedding. The compact-supportedness of the homotopies in [11, Theorems 2.2 and 2.3] does not, by itself, imply the relative h-principle with a non-trivial convex end; one must show that the formal solution can be chosen so that its boundary value at the positive end is the prescribed Legendrian embedding, and that the h-principle can be run relative to both ends. As written, the paper gives no proof of this adaptation and no reference for it, so Theorem 1 does not follow from the cited results.
[§2, paragraph beginning 'For simplicity'] Even granting the extension in Remark 5, the paper does not verify the formal Lagrangian data required to apply the h-principle to the smooth concordance. The sentence 'the complexiﬁed tangent bundle of a concordance over S2n is trivial' does not by itself establish the existence of a Lagrangian monomorphism TL → T(R×R^{4n+1}) covering the embedding, homotopic to the differential, and restricting to the Legendrian differentials at the two ends. The author should either spell out this formal data (for example, by exhibiting a Lagrangian subbundle and a homotopy) or cite the standard h-principle that guarantees it; otherwise the application of the h-principle is incomplete.

minor comments (4)

[§2, paragraph after Remark 4] The sentence 'From [15, Proposition A.4 (c)] it follows that for a fixed rotation class, there is exactly one such couple Λ−, Λ+ up to Legendrian isotopy' is confusingly phrased; Remark 6 clarifies that for even k > 2 there are two Legendrian non-isotopic embeddings with the same classical invariants. Please rephrase to avoid implying uniqueness of the pair.
[§2, Remark 6] There is a typo 'Note that the the proof' with a doubled article.
[References] The arXiv number in reference [16] appears as '22105.02390'; this is likely a typo and should read '2210.02390' or another correct identifier.
[§3, last paragraph] The notation 'Si1 × · · ·×Sik × S2' is not defined; please clarify that Si denotes a sphere of dimension i, and explain the role of the indices i1,...,ik.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is derived from external classification, embedding, and h-principle results; the only self-citations are prior published constructions in the auxiliary Section 3.

full rationale

The derivation of Theorem 1 is a chain of external results: Murphy's classification [15, Proposition A.4(c)] provides two loose Legendrian embeddings with the same rotation class that are not Legendrian isotopic; Wu's theorem [18] provides smooth concordances between the underlying smooth embeddings; and the Eliashberg–Murphy h-principle [11, Theorem 2.2] (as adapted in Remark 5) upgrades those smooth concordances to exact Lagrangian concordances. The non-isotopy of the endpoints is an input from Murphy's classification, not a consequence of the constructed concordances, and no equation or parameter is fitted to the target conclusion. Remark 5 asserts an unproven extension of the Eliashberg–Murphy theorem from exact Lagrangian caps to cobordisms with a loose concave end and a possibly non-trivial convex end; this is a genuine correctness risk, because the adaptation is not proved in the paper, but it is not circularity—the paper does not define the h-principle in terms of Theorem 1, and the cited theorem is external. The only self-citations occur in Section 3, where the pair of exact Lagrangian cobordisms in dimension three is taken from the author's earlier published paper [7, Section 2.3] and then extended to high dimensions by front spinning; citing a previously published construction is not circular, and this section is auxiliary rather than the basis of Theorem 1. No fitted-input-called-prediction, no self-definitional step, and no renaming of a known result as new were found. Therefore the paper does not exhibit significant circularity; the main open issue is the correctness of the unproven adaptation in Remark 5, which is a mathematical validity concern rather than a circularity concern.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof rests on deep external theorems rather than new postulates. There are no fitted parameters and no invented entities. The load-bearing input is Murphy's loose Legendrian classification, and the key unproved step is the adaptation of the Eliashberg-Murphy h-principle in Remark 5, which is the main reason the soundness score is not higher.

assumptions (5)

domain assumption Murphy's loose Legendrian classification in [15, Prop A.4 and Theorem 1.2]
Provides the non-isotopic pair of loose Legendrian embeddings with the same rotation class, the starting point of the construction in Section 2. The paper relies on the specific reading of the classification for even k > 2.
standard math Wu's theorem that every two embeddings of a connected 2n-manifold into R^{4n+1} are smoothly isotopic for n ≥ 1
Used in Section 2 to obtain smooth concordances between the two Legendrian embeddings, giving the manifolds LC∞± and LC∞∓ that are later upgraded to Lagrangian concordances.
domain assumption Adaptation of the Eliashberg-Murphy h-principle, [11, Theorem 2.2], to exact Lagrangian cobordisms with loose concave ends and possibly non-trivial convex ends
This step upgrades the smooth concordances to Lagrangian concordances in the proof of Theorem 1. The adaptation is asserted in Remark 5 but not proven in the paper, making it the main load-bearing unproved assumption.
standard math The complexified tangent bundle of a concordance over S^{2n} is trivial
Invoked in Section 2 to ensure the formal Lagrangian data for the h-principle exists; follows from the stable parallelizability of spheres.
domain assumption Loose Legendrian submanifolds have acyclic Legendrian contact homology
Used in Remark 4 to assert that Λ− and Λ+ have the same Legendrian invariants. This is a known result about loose Legendrians and is not load-bearing for Theorem 1.

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Cite this review

Pith. "Pith review of Lagrangian concordance is not a partial order in high dimensions." pith.science (2026). https://pith.science/paper/OFPQQCWP

@misc{pith2026241112114,
  author       = {Pith},
  title        = {Pith review of: Lagrangian concordance is not a partial order in high dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OFPQQCWP}},
  note         = {Machine review of arXiv:2411.12114}
}

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, $n>1$.

Figures

Figures reproduced from arXiv: 2411.12114 by the authors.

**Figure 1.** The pair of exact Lagrangian cobordisms L S 2 T2 (left) and L T 2 S2 (right) from [7, Section 2.3]. This pair of cobordisms L T 2 S2 , L S 2 T2 contradicts the anti-symmetry property by the obvious topological reason, i.e. by the fact that S 2 is not diffeomorphic to T 2 . This example can easily be extended to high dimensions by using the front spinning construction (or the spherical spinning construction) applied … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 17 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [8]

Instability of Legendrian knottedness, and non-regular Lagrangian concordances of knots

G. Dimitroglou Rizell and R. Golovko, Instability of Legendrian knottedness, and non-regular La grangian concordances of knots , preprint 2024, available at arXiv:2409.00290. LAGRANGIAN CONCORDANCE IS NOT A PARTIAL ORDER IN HIGH DIMENS IONS 5

work page Pith review arXiv 2024

[2] [1]

Bourgeois, J

F. Bourgeois, J. Sabloﬀ and L. Traynor, Lagrangian cobordisms via generating families: construct ion and geography, Algebraic and Geometric Topology 15 (2015) 2439–2477

work page 2015

[3] [2]

Chantraine, On Lagrangian concordance of Legendrian knots , Algebr

B. Chantraine, On Lagrangian concordance of Legendrian knots , Algebr. Geom. Topol. 10 (2010), 63–85

work page 2010

[4] [3]

Chantraine, Lagrangian concordance is not a symmetric relation , Quantum Topol

B. Chantraine, Lagrangian concordance is not a symmetric relation , Quantum Topol. 6 (2015), no. 3, 451–474

work page 2015

[5] [4]

Chantraine, G

B. Chantraine, G. Dimitroglou Rizell, P. Ghiggini and R. Golovko, Floer theory for Lagrangian cobor- disms, Journal of Diﬀerential Geometry, Volume 114, Number 3 (2020), 393–465

work page 2020

[6] [5]

Chantraine, G

B. Chantraine, G. Dimitroglou Rizell, P. Ghiggini and R. Golovko, Floer homology and Lagrangian concordance, Proceedings of the G¨ okova Geometry-Topology Conference 20 14, (2015), 76–113

work page 2015

[7] [6]

Cornwell, L

C. Cornwell, L. Ng, and S. Sivek, Obstructions to Lagrangian concordance , Algebr. Geom. Topol. 16 (2016), no. 2, 797–824

work page 2016

[8] [7]

Dimitroglou Rizell and R

G. Dimitroglou Rizell and R. Golovko, On homological rigidity and ﬂexibility of exact Lagrangian endo- cobordisms, International Journal of Mathematics, Vol. 25, No. 10, 145009 8 (2014), 24 pages

work page 2014