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Jiggling: an h-principle without homotopical assumptions

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Pith's one-line read Every continuous section can be jiggled into a piecewise smooth solution of any open and fiberwise dense first-order differential relation.

desk verdict A genuinely new generalization of Thurston's jiggling with a solid main theorem, but the advertised simplicial-set h-principle has a gap in the write-up that needs patching. read the letter →

arxiv 2501.13627 v2 pith:WHT3ZXWT submitted 2025-01-23 math.GT math.DG

Anna Fokma , Álvaro del Pino , Lauran Toussaint This is my paper · ORCID
classification math.GTmath.DG MSC 57R4557Q6557R0557R35
keywords h-principlejigglingpiecewisesmoothsectionsdifferentialrelationscrystallinesubdivisionweakhomotopyequivalencecontactformsgeneralposition
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 claims a broad generalization of the classical jiggling lemma: for any open and fiberwise dense first-order differential relation $R \subset J^1(E)$, every piecewise smooth section of the bundle $E$ can be subdivided and then $C^1$-perturbed into a piecewise smooth solution of $R$, while staying fixed on any region where it already solves $R$. The construction is parametric and relative, and it yields an h-principle without homotopical assumptions: the simplicial set of piecewise smooth solutions is weakly equivalent to the singular complex of all continuous sections. This matters because one mechanism then covers many geometric flexibility statements at once — transversality, immersions, submersions, piecewise contact forms, and general-position triangulations — on closed manifolds as well as open ones, where classical smooth h-principles often require formal data or fail.

What carries the argument

The engine is the jiggling construction of Section 6, which combines three ingredients. Crystalline subdivision refines a triangulation by pulling back the standard subdivision of a cube; its defining feature is that every simplex in every iterated subdivision is a scaled, translated copy of one of finitely many model simplices, so the quantities controlling $C^1$ estimates (edge length and the degeneracy constant $\Lambda$) scale predictably with subdivision depth. A fixed finite coloring of the top-dimensional simplices allows the proof to perturb the section one color at a time, because same-colored simplices have disjoint stars and earlier solutions are only mildly disturbed. On each simplex, slope perturbation uses fiberwise density to tilt the jet of the section into the relation and then extends the tilted jet linearly, while a join construction glues the local tilts to the part of the section that is left unchanged. The parametric statement is encoded in the simplicial set $sSol_{PS}(R)$, whose $n$-simplices are piecewise solutions of the pulled-back relation on $M \times \Delta^n$.

What would settle it

Consider the immersion relation on maps $M \to N$ with $\dim M < \dim N$, which is open and fiberwise dense, and a loop in $C^0(M,N)$ whose values are immersions; Theorem 7.4 predicts a piecewise immersive filling of that loop. A loop for which one can prove that no piecewise smooth immersion filling exists would falsify the central weak equivalence.

Watch

Extended reading notes

Core claim

The central discovery is that jiggling is not just a way to put triangulations in general position: it is a way to solve differential relations. Given an open and fiberwise dense relation $R \subset J^1(E)$, Theorem 6.3 takes any piecewise smooth section $s$ and any positive error function $\varepsilon$ and produces an $\varepsilon$-jiggling $(s', T')$ in which $s'$ is a piecewise smooth solution of $R$ and $s'$ agrees with $s$ on any prescribed subcomplex $Q$ where $s$ already solves $R$. Because the proof is relative and uniform over parameters, Theorem 7.4 follows: the inclusion of the simplicial set $sSol_{PS}(R)$ of piecewise solutions into $\operatorname{Sing}(\Gamma^0(E))$ is a weak homotopy equivalence of Kan complexes, meaning simplicial sets in which every horn can be filled. The authors read this as an h-principle without homotopical assumptions, since the first jet of a piecewise solution may jump along the triangulation and hence no formal solution or jet homotopy class appears in the statement.

Load-bearing premise

The load-bearing premise is fiberwise density: at every point of the bundle, the allowed first-order jets must be dense among all possible slopes over that point, so that any jet can be tilted into the relation by an arbitrarily small change of slope.

Editorial extensions

If this is right

  • Every continuous section of $E$, and every continuous family of sections parametrized by a simplex, can be deformed relatively into a piecewise smooth solution of $R$; in simplicial terms, $sSol_{PS}(R)$ and $\operatorname{Sing}(\Gamma^0(E))$ have the same weak homotopy type.
  • Any map between manifolds can be jiggled to be piecewise transverse to a fixed distribution, and, in the appropriate dimensions, to be piecewise immersive or submersive; the corresponding inclusions of solution spaces into spaces of continuous maps are weak homotopy equivalences.
  • On any odd-dimensional manifold, every nonvanishing 1-form is $C^1$-close to a piecewise smooth contact form, and the simplicial set of piecewise contact forms is weakly equivalent to the space of nonvanishing continuous 1-forms, on closed manifolds as well as open ones.
  • Every triangulation can be jiggled to be in very general position with respect to a distribution, including on noncompact manifolds, and the same statement holds for finitely many distributions at once.

Reading between the lines

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

  • Editorial inference: if Theorem 7.4 is correct, then in the piecewise smooth category every open and fiberwise dense first-order relation is automatically flexible at the level of homotopy types; rigidity phenomena, such as the conjectured rigidity of positive contact structures in dimension 3, would have to come precisely from failure of fiberwise density.
  • Editorial inference: the color-by-color induction is the transferable core of the proof, since it replaces delicate Grassmannian distance estimates with a finite coloring; a testable extension is to higher-order jets, where a subdivision scheme controlling higher derivatives might yield analogous h-principles without homotopical assumptions for relations in $J^r(E)$.
  • Editorial inference: one could try to replace fiberwise density by an ampleness condition in the sense of convex integration; if sufficiently controlled local tilts exist without density, jiggling would extend piecewise h-principles to a larger class of relations, including some that are not fiberwise dense.
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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 / 3 minor

Summary. The paper proves a generalized jiggling theorem (Theorem 6.3): for any open and fiberwise dense differential relation R ⊂ J^1(E) of first order, every piecewise smooth section of E can be subdivided and C^1-approximated by a piecewise smooth solution of R, with relative and non-compact variants. The proof works by choosing a fine crystalline subdivision, linearizing the section, and then slope-perturbing it color by color, using quantitative estimates on the geometry of crystalline subdivisions. The authors then package this into an h-principle without homotopical assumptions: the simplicial set sSol_PS(R) of piecewise smooth solutions over M × Δ^n is weakly equivalent to the singular complex of the space of continuous sections of E (Theorem 7.4). Applications are given to piecewise transversality, immersions and submersions, contact forms, and a new proof of Thurston's jiggling lemma, including non-compact manifolds.

Significance. If Theorem 6.3 and Theorem 7.4 are correct, this is a substantial contribution to the flexibility of piecewise smooth first-order differential relations. The paper gives a self-contained, quantitative proof of the core jiggling result, with explicit bounds coming from model simplices (Lemma 3.8), linearization (Lemma 4.2), interpolation and join estimates (Lemmas 4.4 and 4.6), and slope-perturbation estimates (Lemma 5.8). The color-by-color induction in Theorem 6.1 is a genuine simplification of Thurston's original argument and is carefully set up. The applications to contact forms and to general position are attractive and show the power of the result. There is no circularity or fitting of parameters in the main construction. The central weakness is in Section 7: the passage from jiggling to the simplicial h-principle contains an unproved relative piecewise-smooth extension/approximation step. This step is load-bearing for both Lemma 7.3 and Theorem 7.4, so the paper needs a dedicated argument there before the announced h-principle is fully established.

major comments (2)
  1. [Section 7 (Lemma 7.3)] The proof of Lemma 7.3 states: 'Extend s to a piecewise smooth section tilde{s} of E over M × Δ^n such that it is piecewise smooth with respect to a triangulation tilde{T} that extends T. We assume that tilde{s} is a solution on Op(M × Λ^n_k).' The second sentence is not automatic from openness of R. A solution on the horn fixes the values and derivatives on the horn, but it gives no control on the normal derivatives used to extend into the missing faces; for n ≥ 2 the collars of the horn meet in corners, so the extension has to be assembled simplex by simplex and across faces. Because the Kan property of sSol_PS(R) is asserted from this lemma and is then used in the weak-equivalence criterion (Definition 2.29), this gap must be closed by a proof or by a precise reference to a relative extension theorem for piecewise smooth sections of open relations.
  2. [Section 7 (proof of Theorem 7.4)] The proof begins with the assertion: 'Firstly, we approximate F by a section F′ that is piecewise smooth such that F′ is a solution on Op(M × ∂Δ^n) and such that F′ agrees with F on M × ∂Δ^n.' No argument is supplied for this step. This is load-bearing: Theorem 6.3 can only be applied to a piecewise smooth F′ that is already a solution on a neighborhood of the relative subcomplex Q = M × ∂Δ^n. The requirement F′ = F on the boundary is an exact interpolation condition, while the requirement that first-order jets in the collar lie in R is a differential condition; openness of R alone does not imply that an arbitrary piecewise-smooth approximation of F satisfies the latter. The same type of unproved extension appears in Lemma 7.3. A dedicated relative approximation/extension lemma for piecewise smooth sections over products M × Δ^n, with corners, is needed here.
minor comments (3)
  1. [Section 6.1 (proof of Theorem 6.1)] In the paragraph handling simplices of a previous color, the inequality dC0(j1s(i), j1s(i+1)) < δ_i/2 does not follow from the stated item (1), which gives a δ_i/4 bound for an ε_{i+1}-slope perturbation on adjacent simplices. The indices and constants should be harmonized (probably δ_i/4), although this is a local fix and does not affect the overall strategy.
  2. [Section 2.2.5 (Definition 2.12)] The phrasing 'a triangulation K′ subdividing K1, K2, K' should be 'subdividing K1, K2, and K'. Also, the fact that the C^r distance depends on the auxiliary triangulation K is only explained after the definition; putting that caveat before the displayed formula would avoid confusion.
  3. [Section 7.1] The informal discussion of the topological-space analogue is useful motivation, but it contains the phrase 'we leave for the reader to explore'; if this is not intended as a mathematical claim, it is fine as a remark, but it should be clearly marked as non-essential and not as a lemma.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the jiggling construction is self-contained; the only flagged issue is an unproved relative approximation step in Theorem 7.4, which is a proof gap rather than a circular reduction.

full rationale

The central derivation is self-contained. Theorem 6.3 is proved by an explicit construction: linearize relative to a crystalline subdivision (Corollary 4.7, Lemma 4.2), color simplices (Lemma 3.3), slope-perturb over colors using Lemma 5.1 and Corollary 5.5, and glue with joins and interpolations (Lemmas 4.4, 4.6, 5.8). No parameter is fitted to the target conclusion: the constants epsilon_i, delta_i and l are chosen from the relation R and the input section s, not from the existence of a solution. The only self-citations ([6], [26]) appear in the introduction and applications as illustrations of prior uses of jiggling and Engel structures; they are not load-bearing for Theorem 6.3 or Theorem 7.4. The h-principle Theorem 7.4 reduces to Theorem 6.3, plus standard simplicial-set facts and the contractibility of the space of triangulations [21], which is external. I flag one omitted proof, as required by the review rules: in the proof of Theorem 7.4 the paper asserts 'Firstly, we approximate F by a section F′ that is piecewise smooth such that F′ is a solution on Op(M × ∂∆n) and such that F′ agrees with F on M × ∂∆n' without proving this relative piecewise-smooth approximation and extension step; the same kind of extension is asserted in Lemma 7.3. This is a correctness and completeness gap, likely fillable by standard relative smooth approximation plus openness of R, but it is not a circularity, because nothing in that step is defined in terms of the conclusion or fitted to it. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof is a self-contained geometric construction. No numbers are fitted to data and no new mathematical entities are postulated. The only substantive input besides standard background is the openness and fiberwise density of the relation R, which is an explicit hypothesis rather than a hidden assumption.

assumptions (4)
  • standard math Every smooth manifold admits a smooth triangulation compatible with its smooth structure (Whitehead).
    Used throughout; see Definition 2.19 and the proof of Theorem 6.3, citing [33].
  • standard math Any triangulation of M×∂∆^n extends to a triangulation of M×∆^n; the space of triangulations is contractible.
    Used in Lemma 7.3 and Theorem 7.4 to extend piecewise smooth data, citing [21].
  • standard math The singular complex Sing(X) is a Kan complex and Definition 2.29 correctly characterizes weak homotopy equivalences of Kan complexes.
    Background for Section 7.2; cited [16, 18, 10].
  • domain assumption The relation R is open and fiberwise dense in J^1(E).
    This is the key hypothesis of Theorem 6.3; it enters through Lemma 5.1 and Corollary 5.5 to produce local solutions.

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Pith. "Pith review of Jiggling: an h-principle without homotopical assumptions." pith.science (2026). https://pith.science/paper/WHT3ZXWT

@misc{pith2026250113627,
  author       = {Pith},
  title        = {Pith review of: Jiggling: an h-principle without homotopical assumptions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WHT3ZXWT}},
  note         = {Machine review of arXiv:2501.13627}
}
abstract

The jiggling lemma of Thurston shows that any triangulation can be jiggled (read: subdivided and then perturbed) to be in general position with respect to a distribution. Our main result is a generalization of Thurston's lemma. It states that piecewise smooth solutions of a given open and fiberwise dense differential relation $\mathcal{R} \subset J^1(E)$ of first order can be constructed by jiggling arbitrary sections of $E$. Our statement also holds in parametric and relative form. We understand this as an h-principle without homotopical assumptions for piecewise smooth solutions of $\mathcal{R}$.

Figures

Figures reproduced from arXiv: 2501.13627 by the authors.

Figure 1
Figure 1. The second crystalline subdivision of a 2-simplex ∆. Definition 3.1. Applying crystalline subdivision to an (ordered) linear m-simplex ∆ with m > 0 is done using the following steps: (1) Include ∆ = ⟨vi0 , . . . , vim⟩ into the standard m-cube I m via the piecewise linear map ι defined by ι(vij ) = ( j z }| { 0, . . . , 0, m−j z }| { 1, . . . , 1). (2) Subdivide I m into 2m smaller m-cubes of size 1/2. (3) Subdivide… view at source ↗
Figure 2
Figure 2. We illustrate here subcomplexes K′ and K′′ of a simplicial complex K, where K′ is not nice and K′′ is. We have illustrated Definition 3.9 in [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. The subcomplex K \ star(K′ ) of K is subdivided in a crystalline manner, but K′ not. Hence we apply the barycentric cone off within ring(K′ ) with respect to an arbitrary crystalline subdivision of K \ star(K′ ). To address this, we must work relatively. Concretely, we need a scheme capable of subdividing the simplicial complex K over a given region, while keeping it the same elsewhere. This will allow us to keep su… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: A generalized crystalline subdivision of a polyhedron K. We have indi￾cated below the crystalline subdivisions of K and in-between we use the barycentric cone off which we have indicated in a lighter color. Proposition 3.14. Let K be a simplicial complex in R N , let δ…
Figure 5
Figure 5. Figure 5: An example of the result s ′ of slope-perturbing s over a simplex ∆ in K. We use the join over star(∆1 ). The following statement is immediate from the bounds established in Lemmas 4.6 and 5.6: Lemma 5.8. Let K be a colored simplicial complex of pure dimension m. Let K…
Figure 6
Figure 6. Figure 6: A sketch of the proof of Theorem 6.3 in the non-compact case when about to construct s (k) . The compacts Ck−1 and Ck lie left of the respective orange lines. We have drawn K(k−1) and indicated the subcomplex L (k) and its stars. On top we indicate how to obtain s (k) …
Figure 7
Figure 7. Figure 7: We give two examples of a distribution ξ on ∆2 such that ∆2 is (not) in general position with respect to ξ. In the following definition we identify a top-dimensional linear simplex ∆ in a simplicial complex K with the standard simplex ∆m ⊂ R m. We also use the fact tha…
Figure 8
Figure 8. Figure 8: Example of a triangulation in general position with respect to the vertical distribution ξ whose subdivision is not. 8.3.2. Jiggling. We now fit general position into our jiggling framework. We are given a map f : M → N that is piecewise smooth with respect to a triang…

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  1. Thurston's jiggling

    math.GT 2025-08 conditional novelty 5.0 of 10

    A new proof of Thurston's jiggling lemma, including a full proof of the manifold case using relative jiggling and quantitative semitransversality.

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Works this paper leans on

34 extracted references · 24 canonical work pages · cited by 1 Pith paper

  1. [1]

    Arboreal models and their stability

    D. ´Alvarez-Gavela, Y. Eliashberg, and D. Nadler. “Arboreal models and their stability”. In: Journal of Symplectic Geometry 21.2 (2023), pp. 331–381

    work page 2023

  2. [2]

    Geomorphology of Lagrangian ridges

    D. ´Alvarez-Gavela, Y. Eliashberg, and D. Nadler. “Geomorphology of Lagrangian ridges”. In: Journal of Topology 15.2 (2022), pp. 844–877

    work page 2022

  3. [3]

    ´Alvarez-Gavela, Y

    D. ´Alvarez-Gavela, Y. Eliashberg, and D. Nadler.Positive arborealization of polarized Weinstein manifolds. 2022. arXiv: 2011.08962 [math.SG]

    arXiv 2022

  4. [4]

    Entrelacements et ´ equations de Pfaff

    D. Bennequin. “Entrelacements et ´ equations de Pfaff”. In: Asterisque 107–108 (1983), pp. 87– 161

    work page 1983

  5. [5]

    Bertelson and J

    M. Bertelson and J. Distexhe. PL approximations of symplectic manifolds . 2024. arXiv: 2112. 10118 [math.DG]

    2024

  6. [6]

    Existence h-Principle for Engel structures

    R. Casals, J.L. P´ erez, A. del Pino, and F Presas. “Existence h-Principle for Engel structures”. In: Invent. Math. 210 (2017), pp. 417–451

    work page 2017

  7. [7]

    Cieliebak, Y

    K. Cieliebak, Y. Eliashberg, and N. Mishachev. Introduction to the h-principle. Vol. 239. Grad- uate Studies in Mathematics. American Mathematical Society, Providence, RI, 2023

    work page 2023

  8. [8]

    Stabilit´ e topologique des structures de contact en dimension 3

    V. Colin. “Stabilit´ e topologique des structures de contact en dimension 3”. In: Duke Mathemat- ical Journal 99.2 (1999), pp. 329–351

    1999

Show all 34 references
  1. [9]

    Symplectic submanifolds and almost-complex geometry

    S.K. Donaldson. “Symplectic submanifolds and almost-complex geometry”. In: J. Differential Geom. 44.4 (1996), pp. 666–705

    1996

  2. [10]

    Weak equivalences of simplicial presheaves

    D. Dugger and D. C. Isaksen. “Weak equivalences of simplicial presheaves”. In: Contemporary Mathematics 346 (2004), pp. 97–114

    2004

  3. [11]

    A theorem on the structure of wave fronts and its applications in symplectic topology

    Y. Eliashberg. “A theorem on the structure of wave fronts and its applications in symplectic topology”. In: Functional Analysis and Its Applications 21.3 (1987), pp. 227–232

    1987

  4. [12]

    Classification of overtwisted contact structures on 3-manifolds

    Y. Eliashberg. “Classification of overtwisted contact structures on 3-manifolds”. In: Invent. Math. 98.3 (1989), pp. 623–637

    1989

  5. [13]

    Contact 3–manifolds twenty years since J. Martinet’s work

    Y. Eliashberg. “Contact 3–manifolds twenty years since J. Martinet’s work”. In: Ann. Inst. Fourier 42 (1992), pp. 165–192

    1992

  6. [14]

    M. Gromov. Partial differential relations . Vol. 9. Ergebnisse der Mathematik und ihrer Gren- zgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1986, pp. x+363

    1986

  7. [15]

    Stable mappings of foliations into manifolds

    M. Gromov. “Stable mappings of foliations into manifolds”. In: Mathematics of the USSR- Izvestiya 3.4 (1969), p. 671

    1969

  8. [16]

    Heuts and I

    G. Heuts and I. Moerdijk. Simplicial and dendroidal homotopy theory . Springer Nature, 2022

    2022

  9. [17]

    Coisotropic rigidity andC 0-symplectic geometry

    V. Humili` ere, R. Leclercq, and S. Seyfaddini. “Coisotropic rigidity andC 0-symplectic geometry”. In: Duke Mathematical Journal 164.4 (2015), pp. 767–799

    2015

  10. [18]

    Simplical presheaves

    J. Jardine. “Simplical presheaves”. In: Journal of Pure and Applied Algebra 47.1 (1987), pp. 35– 87

    1987

  11. [19]

    No symplectic Lipschitz structures on S2n≥4

    D. Joksimovi´ c. “No symplectic Lipschitz structures on S2n≥4”. In: International Journal of Mathematics 33.05 (2022), pp. 1–10

    2022

  12. [20]

    Haefliger structures and symplectic/contact structures

    F. Laudenbach and G. Meigniez. “Haefliger structures and symplectic/contact structures”. In: Journal de l’ ´Ecole polytechnique - Math´ ematiques3 (2016), pp. 1–19

    2016

  13. [21]

    J. Lurie. Topics in geometric topology. https://www.math.ias.edu/~lurie/937.html. 2009

    2009

  14. [22]

    Sur l’existence de certaines formes diff´ erentielles remarquables sur la sph` ere S3

    R. Lutz. “Sur l’existence de certaines formes diff´ erentielles remarquables sur la sph` ere S3”. In: C.R. Acad. Sci. Paris 270 (1970), pp. 1597–1599

    1970

  15. [23]

    Formes de contact sur les vari´ et´ es de dimension 3

    J. Martinet. “Formes de contact sur les vari´ et´ es de dimension 3”. In:Proc. Liverpool Sing. Symp. II, Lect. Notes in Math. 209 (1971), pp. 142–163

    1971

  16. [24]

    Gromov’s alternative, Eliashberg’s shape invariant, andC 0-rigidity of contact diffeomorphisms

    S. M¨ uller and P. Spaeth. “Gromov’s alternative, Eliashberg’s shape invariant, andC 0-rigidity of contact diffeomorphisms”. In: International Journal of Mathematics 25.14 (2014), p. 1450124

    2014

  17. [25]

    C0-rigidity of characteristics in symplectic geometry

    E. Opshtein. “ C0-rigidity of characteristics in symplectic geometry”. In: Annales scientifiques de l’Ecole normale sup´ erieure42.5 (2009), pp. 857–864

    2009

  18. [26]

    The Engel–Lutz twist and overtwisted Engel structures

    ´A. del Pino and T. Vogel. “The Engel–Lutz twist and overtwisted Engel structures”. In: Geom- etry & Topology 24.5 (2020), pp. 2471–2546

    2020

  19. [27]

    Chekanov’s dichotomy in contact topology

    D. Rosen and J. Zhang. “Chekanov’s dichotomy in contact topology”. In: Mathematical Research Letters 27.4 (2020), pp. 1165–1193. REFERENCES 35

    2020

  20. [28]

    ∆-sets I: Homotopy theory

    C.P. Rourke and B.J. Sanderson. “∆-sets I: Homotopy theory”. In: The Quarterly Journal of Mathematics 22.3 (1971), pp. 321–338

    1971

  21. [29]

    The Seiberg-Witten invariants and symplectic forms

    C. H. Taubes. “The Seiberg-Witten invariants and symplectic forms”. In: Mathematical Research Letters 1.6 (1994), pp. 809–822

    1994

  22. [30]

    Existence of codimension-one foliations

    W.P. Thurston. “Existence of codimension-one foliations”. In: Ann. Math. 104.2 (1976), pp. 249– 268

    1976

  23. [31]

    The theory of foliations of codimension greater than one

    W.P. Thurston. “The theory of foliations of codimension greater than one”. In: Comm. Math. Helv. 87 (1974), pp. 214–231

    1974

  24. [32]

    On the uniqueness of the contact structure approximating a foliation

    T. Vogel. “On the uniqueness of the contact structure approximating a foliation”. In: Geometry & Topology 20.5 (2016), pp. 2439–2573

    2016

  25. [33]

    On C 1-complexes

    J.H.C. Whitehead. “On C 1-complexes”. In: Ann. of Math. 41.2 (1940), pp. 809–824

    1940

  26. [34]

    H. Whitney. Geometric Integration Theory. Princeton University Press, 1957. Utrecht University, Department of Mathematics, Budapestlaan 6, 3584 CD Utrecht, The Netherlands Email address : annaf.math@outlook.com Utrecht University, Department of Mathematics, Budapestlaan 6, 358...

    1957

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