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The h-principle fails for prelegendrians in corank 2 fat distributions

T0 review · 2 major / 7 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For corank-2 fat distributions, formal prelegendrian classes split: infinitely many tori are formally equivalent yet pairwise non-isotopic.

desk verdict First rigidity for prelegendrians in fat distributions, likely correct, but the formal-class preservation step in Prop. 9.1 needs a full proof. read the letter →

arxiv 2511.17780 v2 pith:6GVDYZZW submitted 2025-11-21 math.SG math.DGmath.GT

Eduardo Fernández , Álvaro del Pino , Wei Zhou This is my paper · ORCID
classification math.SGmath.DGmath.GT MSC 53D1053D1253D3557R1758A30
keywords h-principlefatdistributionsprelegendrianscorank2Legendriancontacthomologyco-realsubmanifoldsfrontspinningcontactisation
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

The paper introduces prelegendrian submanifolds in corank-2 fat distributions—maximally non-integrable even-rank distributions that generalize contact structures—and proves that their classification does not reduce to formal, bundle-theoretic data. In the standard fat space C^{2n+1}, for every n≥1, there are infinitely many prelegendrian (2n+1)-tori that all represent the same formal prelegendrian class but are pairwise not prelegendrian isotopic. The obstructions are detected by Legendrian contact homology of their canonical Legendrian lifts to the contactisation, so the failure of the h-principle is genuinely geometric. The same front-based machinery yields a prelegendrian stabilization whose lift is loose, and extends the constructions to arbitrary corank-2 fat distributions in dimension 6. This would be the first example of rigidity for maximally non-integrable distributions beyond contact topology.

What carries the argument

The load-bearing mechanism is the prelegendrian front calculus. Theorem 8.1 characterizes simple prelegendrian fronts: a simple front F ⊆ (X, J) is a prelegendrian front if and only if its singularity loci are co-real in the almost complex manifold (X, J). Prelegendrian spinning along a co-real embedding T^{2n} → C^{n+1} converts 1-dimensional Legendrian knots into prelegendrian (2n+1)-tori, and the canonical Legendrian lift of the result is exactly the Legendrian spinning of the input knot. Legendrian contact homology of that lift therefore becomes a prelegendrian invariant. Formal equivalence among the spun tori is produced by prelegendrian Reidemeister moves and prelegendrian N-pushing al

What would settle it

Exhibit a prelegendrian isotopy between Λ_s and Λ_t for two distinct indices, or a Legendrian isotopy between their canonical lifts in the contactisation C(C^{2n+1}, D_std); either would contradict Theorem 1.10. A more computational check is to compute the Legendrian contact homology of the lifts L(Λ_s) and test whether the resulting algebras for distinct s are quasi-isomorphic—a quasi-isomorphism between any pair would break the detection argument.

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Extended reading notes

Core claim

The central discovery is Theorem 1.10: there exist infinitely many embeddings Λ_s : T^{2n+1} → (C^{2n+1}, D_std), s ∈ Z_{>0}, that are formally prelegendrian isotopic but pairwise not prelegendrian isotopic. Their canonical Legendrian lifts L(Λ_s) to the contactisation C(C^{2n+1}, D_std) are pairwise not Legendrian isotopic, distinguished by Legendrian contact homology. Because formal prelegendrian isotopy is the bundle-theoretic/homotopy notion and genuine prelegendrian isotopy is geometric, this is a failure of the h-principle for prelegendrian embeddings in all dimensions. The proof establishes a front criterion—simple prelegendrian fronts are exactly those whose singularity locus is co-r

Load-bearing premise

The load-bearing premise is that the local prelegendrian N-pushing move and the intervening prelegendrian Reidemeister moves preserve the formal prelegendrian embedding class; the proof of this rotation/interpolation step is asserted by analogy with the contact case rather than written out in detail.

Editorial extensions

If this is right

  • In the standard corank-2 fat space, formal prelegendrian isotopy is strictly weaker than genuine prelegendrian isotopy, in every dimension n ≥ 1.
  • Legendrian contact homology of the canonical lift is an effective prelegendrian invariant: it can distinguish prelegendrians that are formally identical.
  • Prelegendrian stabilization produces, from any prelegendrian, another in the same formal class whose Legendrian lift is loose, yielding non-loose examples and exotic pairs including co-normal lifts of hypersurfaces.
  • The rigidity is stable: no compactly supported path of fat structures connects the exotic tori, since such a path would induce a contactomorphism of contactisations and preserve the Legendrian invariants.
  • Every corank-2 fat distribution in dimension 6 contains compact prelegendrians and admits stabilizations; in higher dimensions an analogous statement holds when the local nilpotentised model is the complex Heisenberg algebra.

Reading between the lines

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

  • Editorial inference: the front-and-canonical-lift strategy is a natural template for probing rigidity in corank k > 2 fat distributions; the missing ingredient there is a usable local model, not the invariant itself.
  • Editorial inference: the formal isotopy in the construction passes through fronts with transverse self-intersections, so the proof as written may only eliminate embedded isotopies while leaving the status of prelegendrian immersions open; smoothing those intermediate fronts would strengthen the conclusion.
  • Editorial inference: the stabilization construction suggests a provisional 'loose prelegendrian' class; if such a class obeys an h-principle, the exotic tori would be exactly the non-loose part, mirroring the tight/overtwisted split in contact topology.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 7 minor

Summary. The paper introduces prelegendrian submanifolds in corank-2 fat distributions, studies their fronts and formal classes, and proves that the h-principle fails for prelegendrian embeddings in the standard fat space (C^{2n+1},D_std). The main theorem constructs infinitely many prelegendrian (2n+1)-tori that are formally prelegendrian isotopic but not prelegendrian isotopic, because their canonical Legendrian lifts to the contactisation are distinguished by Legendrian contact homology. The paper also develops a stabilization operation for prelegendrians, proves robustness of the invariants under perturbations of the fat structure, and extends the constructions to general corank-2 fat manifolds in dimension six and to a subclass in higher dimensions.

Significance. If the proof is completed, this is the first rigidity result for submanifolds in maximally non-integrable distributions beyond the contact case, and it opens a promising new direction in the study of fat distributions. The framework developed here — prelegendrian fronts, co-real singular loci, prelegendrian spinning, and stabilization — is natural and well motivated. The paper makes good use of existing tools (front spinning, Legendrian contact homology, Murphy's h-principle) and includes explicit constructions and concrete models. The structural theorems, in particular the characterisation of simple prelegendrian fronts and the identification of contactisations with spaces of contact elements, are carefully presented and are substantial contributions in their own right.

major comments (2)
  1. [§9.1, Proposition 9.1] The last claim of Proposition 9.1 — that p_N(Λ) is formally prelegendrian isotopic to Λ — is the load-bearing step for the same-formal-class assertion in Theorem 1.10, but it is only sketched. The proof says the argument is 'similar' to Proposition 6.2 and that rotating a non-vanishing complex-tangent vector field Z towards ∂z 'provides a formal rotation of the complex tangencies', after which the contact argument applies. This does not exhibit the required 1-parameter family of monomorphisms F_t covering the interpolating embeddings and does not verify the two algebraic conditions of Definition 2.17(1)(a)–(b) at every time. In particular, condition (b) — isotropy of F_t(TΛ)∩D with respect to the fat curvature form — is not automatic when the vertical (CP^n-fibre) component is rotated; the contact case of Proposition 6.2 only enforces the Legendrian condition, not the additional prelegen
  2. [§11.2, proof of Theorem 1.10] The proof asserts that the prelegendrians Λ_s are formally isotopic because their fronts are connected by prelegendrian Reidemeister moves and N-pushings. The paper does not define 'formal prelegendrian isotopy' explicitly, and the proof in §11.2 does not specify the formal data along each segment of the concatenated homotopy. In particular, it is not shown how the formal isotopy produced by the N-pushing in Proposition 9.1 is glued to the genuine prelegendrian isotopies coming from Reidemeister moves, nor that the resulting path is a path of formal prelegendrian embeddings in the sense required by Definition 2.17(3). A complete proof should either prove that the three prelegendrian Reidemeister moves preserve the formal prelegendrian class (with an explicit formal homotopy) or state and prove a composition lemma for formal prelegendrian isotopies. Without this, the conclusion that all Λ
minor comments (7)
  1. [§8.1, Proposition 8.5] In the proof, 'open coition' should be 'open condition'.
  2. [§11.2, proof of Theorem 1.10] The sentence 'the Legendrian L(Λ_s) are pairwise not Legendrian non isotopic' contains a double negation; it should read 'pairwise not Legendrian isotopic'.
  3. [§2.2, Theorem 2.6] In item (4), 'contact contact submanifold' is a typo; it should be 'contact submanifold'.
  4. [§4.1] The phrase 'an alternative characterisation of of fatness' has an extra 'of'.
  5. [§11.1, Lemma 11.1] The word 'functioriality' should be 'functoriality'.
  6. [Figure 5 caption] The caption says 'The first and fourth arrow indicate prelegendrian RII moves, and the last one several prelegendrian RI moves', but 'the last one' is the fourth arrow. The correspondence between arrows and moves should be clarified.
  7. [Definition 2.17] The definition of a formal preisotropic embedding uses a family F_s with F_0 = df and (f,F_1) formal, but it is not stated whether intermediate F_s are required to satisfy the formal conditions. For clarity and for the notion of formal isotopy used later, this should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the formal classes and separating invariants come from independent inputs, and the load-bearing steps are not reductions of conclusions to premises.

full rationale

The paper's central claim (Theorem 1.10) has two independent ingredients: (1) all exotic tori are formally prelegendrian isotopic, and (2) their Legendrian lifts are pairwise non-isotopic. The formal-class statement rests on Lemma 7.5 (formal isotopy of spun Legendrians with equal rotation numbers) and Theorem 9.2/Proposition 9.1 (prelegendrian N-pushing preserves the formal prelegendrian class). These are proven internally: Lemma 7.5 gives a direct proof using Reidemeister moves and Proposition 6.2, and Proposition 9.1 constructs the required vector field Z using Proposition 3.9 and Lemma 3.10, which are proven from explicit bundle splittings. The non-isotopy statement is imported from [EES05b, Example 4.20], an external computation by different authors of Legendrian contact homology for iterated S^1-spinnings of the Whitehead doubles W_s. No parameter is fitted to the target conclusion, and no result is justified solely by a self-citation. The authors' citation of their own earlier conjecture [MAdP21] is contextual and not load-bearing. The most delicate step, Proposition 9.1's sentence 'The last claim will follow by an argument similar to the one in Proposition 6.2', is an asserted analogy rather than a written-out interpolation, but this is a possible proof gap or correctness risk, not a circular reduction: it does not define the conclusion in terms of the hypothesis or rename a fitted input as a prediction. Under the stated rules, a non-finding is therefore appropriate.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

Pure mathematics: no fitting parameters exist. The central claim rests on external theorems (Murphy, Gromov, EES05b, Bhowmick, Rayner, singularity classification) and one cited classification (6D nilpotent Lie algebras). The genuinely new objects — prelegendrians, prelegendrian fronts, co-real spinning, prelegendrian stabilization — are precise mathematical definitions, not postulated entities, and the conclusions are never used in their own proofs. The heaviest external weight is on [EES05b, Example 4.20] (the LCH computation separating the exotica), which is appropriate use of prior work but means the paper's rigidity conclusion inherits the technical status of that computation.

assumptions (7)
  • standard math Murphy's h-principle for loose Legendrians (Theorem 6.5): two loose Legendrians are isotopic iff formally isotopic.
    External theorem [Mur19]; load-bearing for Theorem 1.12 (L(s(Λ)) is loose) and for the formal classification used in Lemma 7.5.
  • standard math Legendrian contact homology distinguishes the spun Whitehead doubles: the LCHs of spin(W_s) are pairwise non-quasi-isomorphic ([EES05b, Example 4.20]).
    External computation [EES05b] supplying the actual rigidity invariant in Theorem 1.10; the paper does not reproduce it.
  • standard math Gromov's h-principle for co-real embeddings (Theorem 3.6) and the Euler-class vanishing for co-real submanifolds of C^{n+1} (Proposition 3.7, [Wel69]).
    Used in Lemma 3.8 and Proposition 10.1 to construct and perturb co-real submanifolds; cited to Gromov and Wells.
  • domain assumption Classification of nilpotent 6-dimensional Lie algebras: a unique one supports a corank-2 fat distribution, the complex Heisenberg algebra (Corollary 12.2 via [CFS05, Mag86]).
    Load-bearing for Theorem 1.17 (every 6D fat distribution contains compact, stabilizable prelegendrians); the uniqueness deduction is cited, not derived in the text.
  • standard math Front singularity normal forms: simple fronts with A1, A1A1, A2 singularities and their genericity (Definition 5.5, Lemma 5.6), citing [AN01, AGV12, Arn90].
    Underpins Theorem 8.1 (prelegendrian front criterion); standard singularity theory.
  • standard math Bhowmick's h-principle for subcritical preisotropic immersions and prelegendrian immersions of open manifolds (Theorem 2.18).
    External result [Bho22] used for context and open questions (1.22); it shows the paper's rigidity for closed embeddings is compatible with known flexibility.
  • standard math The canonical fat distribution D_can on Gr(X,J) is fat (Prop 4.5), proven via the contactisation identification (Theorem 4.7); holomorphic contact structures are fat (Lemma 4.4, via Lemma 4.2's connecting-isomorphism criterion).
    Foundation for all the examples; the corank-2 fatness criterion (connecting isomorphism has no real eigenvalues) is proven in the text.

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Pith. "Pith review of The h-principle fails for prelegendrians in corank 2 fat distributions." pith.science (2026). https://pith.science/paper/6GVDYZZW

@misc{pith2026251117780,
  author       = {Pith},
  title        = {Pith review of: The h-principle fails for prelegendrians in corank 2 fat distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6GVDYZZW}},
  note         = {Machine review of arXiv:2511.17780}
}
abstract

We investigate the $h$-principle problem for fat distributions. These are maximally non-integrable distributions with natural symplectisations and contactisations, that generalize contact distributions to higher corank. We focus on the corank-$2$ case, where we study a natural class of submanifolds, which we call prelegendrians. Their key feature is that they admit a canonical Legendrian lift to the contactisation. Our main results state that the $h$-principle fails for these submanifolds in all dimensions. To the best of our knowledge, this is the first example of rigidity in the study of maximally non-integrable distributions, outside of contact topology. First, we find an infinite family of $(2n+1)$-tori in the standard fat $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$, with the following two properties: (1) They all represent the same formal prelegendrian class, (2) but they are not prelegendrian isotopic because they are distinguished by pseudoholomorphic curve invariants of their Legendrian lift. Secondly, we define the notion of prelegendrian stabilization in $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$. This allows us to take an arbitrary prelegendrian and produce another one, in the same formal class, whose Legendrian lift is loose. In order to prove these results we also develop the fundamentals of the theory of prelegendrians. This includes: (1) introducing the notion of front projection in $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$, (2) proving that pseudoholomorphic curve invariants are robust under perturbations of the fat structure, allowing us to transport our results to non-standard fat structures, (3) introducing a zooming argument showing that any fat structure in dimension $6$ admits prelegendrians.

Figures

Figures reproduced from arXiv: 2511.17780 by the authors.

Figure 1
Figure 1. Left: the front of the cusp model L 0 cusp. Right: the stabilized front Lloose. Let Q be a (n − 1)-dimensional manifold, and define X = R × Q. Consider 1-jet bundle J 1X = J 1R × T ∗Q, and a Legendrian submanifold Lcusp ⊆ J 1R × T ∗Q defined as: Lcusp := L 0 cusp × Q ⊆ J 1R × T ∗Q where Q ⊆ T ∗Q is the 0-section. Notice that over {x ≤ 0} we have that Lcusp ∩ {x ≤ 0} = (j 1 0R ⊔ j 1 1R ∩ {x ≤ 0}) × Q = j 1 0{x≤0}×Q ⊔… view at source ↗
Figure 2
Figure 2. A schematic depiction of the front spinning of the Legendrian unknot U ⊆ (J 1R, ξstd) along the standard circle S 1 ⊆ R 2 . Proof. Following the notations above we write e : K × R 2 → J 0R n , to denote the tubular neigh￾bourhood embedding of ˆe induced by Fr, in which we identify {k} × R 2 = J 0R. By the h-principle for Legendrian immersions in R 3 , there is a generic homotopy of Legendrian immersions Lt , t ∈ [0,… view at source ↗
Figure 3
Figure 3. From left to right the fronts Fˆ s ⊆ J 0R, s ∈ [0, 1], of the 1-dimensional Reidemeister moves I, II and III. The higher dimensional moves are locally modelled on Fˆ s × R n ⊆ J 0R × R n = J 0R n+1 , s ∈ [0, 1]. homotopy of prelegendrian fronts Fs, s ∈ [0, 1], that describes an RI-move along N. Moreover, since being co-real is an open condition we may achieve that the region in the prelegendrian front F1 in which th… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The Legendrian Ws, where s is the number of crossings in the right part. By Corollary 8.10 there is well-defined infinite family of prelegendrian (2n + 1)-tori: Λs = prespineˆ [Ws] ⊆ (C 2n+1 , Dstd) obtained by prelegendrian spinning. Observe that Rot(Ws) = 0 for all s…
Figure 5
Figure 5. Figure 5: A formal prelegendrian isotopy between of Λs = prespineˆ [Ws] and prespineˆ [U], where U ⊆ (J 1R, ξstd) is the standard Legendrian unknot. The first and fourth arrow indicate prelegendrian RII moves, and the last one several prele￾gendrian RI moves along parallel copie…

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Fat distributions with Reeb directions need not be complex contact

    math.GT 2026-03 reject novelty 8.0 of 10

    There exists a global fat (4,6)-distribution on R^6 that has two Reeb directions but is nowhere diffeomorphic to a complex contact structure.

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Works this paper leans on

2 extracted references · 2 linked inside Pith · cited by 1 Pith paper

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