Thurston's jiggling

Anna Fokma

REVIEW 3 major objections 5 minor 15 references

Thurston's jiggling

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves a relative version of the jiggling lemma: any piecewise embedding of a finite complex into a manifold can be subdivided and C^1-perturbed into general position with respect to a distribution, while the map is fixed on a sub

desk verdict A genuinely cleaner route to Thurston's jiggling, with a real but fixable gap in the relative theorem that is the paper's main new claim. read the letter →

arxiv 2508.18759 v1 pith:TIINQC33 submitted 2025-08-26 math.GT math.DG

Anna Fokma This is my paper · ORCID

classification math.GTmath.DG MSC 57R0557Q65

keywords jigglingtriangulationspiecewisesmoothgeneralpositiontransversalitysemitransversalitycrystallinesubdivisiondistributions

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 establishes the jiggling lemma in full generality. The lemma says that any triangulation of a manifold can be subdivided and slightly perturbed so that every simplex—top-dimensional and lower-dimensional faces alike—is transverse to a given distribution. Thurston introduced this technique in the 1970s and gave a detailed proof in Euclidean space, but only sketched the manifold case. The paper's main theorem, Theorem 8.9, proves a relative version: if the map is already in general position on a subcomplex, the jiggling can be chosen to fix that subcomplex exactly. This gives a complete proof of the manifold statement, replacing the extrinsic embed-and-project approach with an intrinsic chart-by-chart induction.

What carries the argument

The load-bearing mechanism is the projection criterion for semitransversality: a simplex join(p,∆) is transverse to a constant foliation precisely when the projection of p misses the projected affine span of ∆. Iterating this criterion, the author perturbs one vertex at a time while keeping a uniform semitransversality radius δ that depends only on the number of simplices and the scale of the subdivision, not on the configuration. Crystalline subdivisions provide the uniform simplex-shape bounds that make the radius scale correctly, and a chart-by-chart induction combined with jiggling the identity map rather than the map itself handles the manifold case.

What would settle it

A concrete test: in R^3 with the horizontal foliation, take the 2-simplex spanned by the origin and two points in the xy-plane, fix the base edge, and ask how far the top vertex can be perturbed while the simplex stays transverse. The theorem predicts a positive, uniform δ-radius independent of the configuration; an explicit configuration in which the required δ shrinks to zero would falsify Proposition 7.9 and with it Proposition 8.1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 8.9: given a finite simplicial complex K, a manifold N with a distribution ξ, any piecewise embedding f:|K|→N, and a subcomplex A on which (f|_{|A|}, A) is already in general position, there is a γ-jiggling (g,K′) of (f,K) such that (g,K′) is in general position with respect to ξ and g equals f on |A|. The proof separates the map from the simplicial complex, works chart by chart, and uses a new "relative jiggling" step to glue charts without losing transversality. Corollary 8.10 recovers Thurston's original jiggling lemma for triangulations of manifolds.

Load-bearing premise

The argument depends on quantitative bounds taken from the companion paper [6] about how subdivision shrinks simplices and how linearization approximates maps; those bounds are not proved here, and if they fail, the uniform control that makes the induction work is lost.

Editorial extensions

If this is right

Corollary 8.10: any smooth triangulation of a manifold can be made in general position over a given compact set by a C^1-small subdivision-and-perturbation, fixing any subcomplex where transversality already holds.
The relative theorem gives a clean way to glue jiggled local charts, which is why the manifold case can be proved without embedding the manifold into Euclidean space.
The proof yields sequences of jigglings whose transversality is uniformly bounded below, not just single jigglings, so the output is quantitative rather than merely existential.
The technique of jiggling a subdivision by deforming the identity map preserves skeleta; this addresses the fact that general position is not preserved under subdivision, a known obstacle for this kind of argument.

Reading between the lines

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

Because Theorem 8.9 fixes A, a direct corollary not stated in the paper is a boundary-relative jiggling: triangulate a compact manifold with boundary, set A to a neighborhood of the boundary, and jiggle the interior while preserving the already-transverse boundary structure.
The skeleton-preserving deformation of the identity is a reusable gadget: it converts a transverse triangulation into a nearby triangulation subdividing a prescribed refinement, which may simplify other constructions where triangulations must be refined without losing transversality.
One could test the quantitative core numerically on small examples, such as a 2-simplex in R^3 with a fixed edge, by measuring the largest δ-radius achievable in Proposition 7.9 and comparing it with the paper's uniform bound; such a check would expose any hidden dependence of δ on the configuration.

Share X Bluesky LinkedIn Reddit HN

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

The paper is worth a look if you care about the jiggling lemma or semitransversality as a tool. What is actually new is the relative jiggling theorem (8.9) and the semitransversality framework that makes the manifold case tractable. The non-relative proof in Sections 6-7 is the strongest part: the estimates are stated carefully, the induction over vertices is transparent, and the compactness arguments on the Grassmannian are honest. That part deserves credit.

The soft spot is exactly where the stress-test points. Proposition 8.8 applies Corollary 8.7 to a map that is only stratified transverse on |star(A,K)|, while Corollary 8.7 requires stratified transversality on all of K. No localization argument is supplied, and you cannot just restrict to the subcomplex because the conclusion has to produce a triangulation of the whole polyhedron with general position on the fixed region. The same issue recurs in the chart-by-chart induction in Theorem 8.9, where Corollary 8.7 is invoked over a union of stars without checking the global hypothesis. This is not a cosmetic gap: the relative construction is the advertised contribution, and as written the proof does not go through.

There are also two smaller issues worth noting. First, the interpolation line in the proof of Proposition 8.8 reads "h_l|Delta = interpolate(...)" which is presumably a typo for g_l|Delta; the intended argument with Lemma 4.5 is plausible once the subdivision-jiggling step is fixed. Second, the paper leans heavily on the companion paper [6] for crystalline subdivision bounds and linearization estimates; that is acceptable if the companion is available and correct, but it makes the present paper not self-contained.

Who gets value from this? Someone working in foliation theory or h-principle who wants a modern account of jiggling, and anyone who wants the semitransversality toolkit for quantitative transversality statements. The non-relative portion is solid enough to cite as an alternative proof of the Euclidean case. The relative theorem is the right kind of result, but it needs a repaired proof.

Recommendation: send it to peer review, but tell the referees to focus on Proposition 8.8 and the use of Corollary 8.7. This is a conditional accept, not a reject.

Referee Report

3 major / 5 minor

Summary. The paper proposes a conceptual proof of Thurston's jiggling lemma, separating the piecewise map from the simplicial complex. The main theorem (Theorem 8.9) states that for a finite simplicial complex K, a manifold N with distribution ξ, a piecewise embedding f:|K|→N, and a subcomplex A on which f|_A is already in general position, there is a γ-jiggling (g,K') that agrees with f on A and is in general position everywhere. The proof plan is: crystalline subdivision estimates, linearization of piecewise maps, quantitative transversality via δ-semitransversality, one-vertex perturbation results (Section 7), Euclidean jiggling (Proposition 8.1), jiggling of subdivisions (Corollary 8.7), a relative Euclidean statement (Proposition 8.8), and a chart-by-chart induction (Theorem 8.9). The paper recovers Thurston's lemma as Corollary 8.10.

Significance. If correct, the result is a significant contribution: it gives a full jiggling proof for general manifolds, which was previously only sketched, and introduces a relative version useful for h-principle arguments. The paper's main strengths are the clean quantitative framework, the detailed induction in Proposition 8.1, and the uniform ε-transversality statement in Corollary 8.3. The set-up is careful and the reliance on the companion paper [6] is explicit. However, the relative construction, which is the paper's claimed new contribution, has a proof gap in the passage from local stratified transversality to the global hypotheses of the subdivision-jiggling statement; this gap is load-bearing for Theorem 8.9.

major comments (3)

[§8.3, proof of Proposition 8.8] The sentence 'We jiggle the subdivision Kbar of K using Corollary 8.7' applies Corollary 8.7 outside its hypotheses. Corollary 8.7 requires f:|K|→N to be stratified transverse with respect to the whole complex K. Proposition 8.8 only assumes stratified transversality on |star(A,K)|. No localization or extension argument is provided to obtain a global sequence (Tℓ) from a jiggling of the subcomplex star(A,K). In particular, the required uniform bounds on rmin, rmax, Λ and the condition dC0(Tℓ,id)<(rmin/4)2^{-ℓ} over all of K do not follow merely from jiggling one subcomplex. This is not a stylistic point: the subsequent interpolation over ring(A,Tℓ) and the relative induction in Theorem 8.9 depend on having such a sequence defined on all of K with controlled quantitative properties.
[§8.4, proof of Theorem 8.9] The relative case A≠∅ is not proved. The final sentence 'In the relative case where A ≠ 0 we apply jiggling relative to A throughout the proof by treating it as K(−1)' does not supply the missing base case or the needed hypotheses. Proposition 8.8 requires (f,K) to be stratified transverse on |star(A,K)|, whereas Theorem 8.9 assumes only that (f|_A,A) is in general position. General position of A does not imply stratified transversality of the simplices in star(A)\A, and subdivision does not preserve general position. Moreover, in the induction step of the A=∅ case, the same issue appears: after extending f^{(i)} to |K|, the text invokes Corollary 8.7 to jiggle the subdivision \tilde K^{(i)}, but Corollary 8.7 requires f^{(i)} to be stratified transverse with respect to the base complex, which is not established; the text only cites general position of f^{(i-1)} on the previous union. Th
[Sections 3–4] The paper delegates the proofs of several load-bearing quantitative results to the companion paper [6]: Lemma 3.8 (crystalline subdivision bounds), Proposition 4.2 (linearization estimates), Lemmas 4.5–4.8 and Corollary 4.9. These estimates control the scaling behavior in ℓ that is essential to Proposition 8.1 and Corollary 8.3. As submitted, the manuscript is not self-contained. If [6] is not yet accepted or publicly available in final form, the referee cannot verify the main theorem. Please either reproduce the necessary proofs or state explicitly the publication status of [6] and ensure the statements quoted here match [6].

minor comments (5)

[Definition 2.7] The definition of dC1 uses an auxiliary triangulation K of the polyhedron P, but K is not explicitly introduced in the statement. It should be stated that K is the fixed auxiliary triangulation used to define the C1-metric.
[§5.3.1] The chart is stated as a map to R^{d(n-d)}; for Gr(n,k) the correct dimension is k(n-k).
[§8.1, induction step] The set D_i is defined using '⟨v_{j0},…,v_{jd}⟩ ∈ star(v_i)'. For d>1 a d-simplex cannot lie in the star of a vertex; this should be phrased in terms of the vertices being contained in vlink(v_i) or the simplex being incident to v_i.
[Lemma 8.5 and Corollaries 8.6–8.7] The bound dC0<γ/2^ℓ uses ℓ without a prior definition in the statement. It should be made explicit that ℓ is the order of crystalline subdivision used in the construction, or the bound should be stated with a named constant.
[Lemma 3.8] The constant E is listed in the statement but is never used in the displayed estimates. Remove it or explain its role.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the jiggling theorem is proved by an explicit inductive construction, not by assuming its conclusion.

full rationale

The central result, Theorem 8.9, is not obtained by circular reasoning. It is proved by constructing jigglings through a chain of independent lemmas: Proposition 8.8, Corollary 8.7, Corollary 8.3, Proposition 8.1, and the vertex-perturbation results of Section 7. None of these assume the target theorem. The paper does import quantitative machinery from the author's companion paper [6]—crystalline subdivision bounds (Lemma 3.8), linearization estimates (Proposition 4.2, Corollary 4.9)—and explicitly says at Section 1.3 that 'Most of Sections 2 to 4 appears already in [6]' and that proofs are omitted. This is a real reliance on self-citation and a missing-support caveat, but it is not circular: the imported results are parameter-free, do not state the jiggling theorem as an assumption, and are not the claim being derived. The skeptic's objection about Corollary 8.7 being invoked in Proposition 8.8 under only local stratified transversality on star(A) identifies a potential proof gap—an unverified hypothesis match—but a proof gap is not circularity, since no equation or definition reduces the conclusion to the hypotheses. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors, and no known result is merely relabeled. Therefore the circularity score is 0.

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

No free parameters or invented entities. The paper's constants (δ, ε, rmax, C) are chosen small or large through compactness, not fitted to data. The main external dependency is the author's companion paper [6] for technical estimates.

assumptions (5)

domain assumption Smooth manifolds admit Whitehead triangulations (Whitehead's theorem)
Used in Corollary 8.10 to apply jiggling to a manifold M triangulated by T:|K|→M.
domain assumption Crystalline subdivision bounds and linearization estimates from [6] (Lemmas 3.3, 3.5, 3.8, Proposition 4.2, Corollary 4.9)
The paper skips proofs of these quantitative results and cites [6] (same author and coauthors). They provide the uniform scaling and C^1-approximation needed for the induction in Proposition 8.1.
standard math Each simplex embedding f|_∆ extends to an embedding of an open neighborhood Op(|∆|) into N
Used in Corollary 8.6 to define the pulled-back distributions ξ_∆ on Op(|∆|); asserted without proof.
domain assumption Constant rank of the distribution ξ on N
The transversality definitions in Section 5 and all estimates assume ξ is a subbundle of constant rank.
standard math Analytic facts about the Grassmannian: compactness, metric comparison (Lemma 5.16), and measure-zero avoidance (Lemma 7.3)
These justify the uniform semitransversality constants in Section 7.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Thurston's jiggling." pith.science (2026). https://pith.science/paper/TIINQC33

@misc{pith2026250818759,
  author       = {Pith},
  title        = {Pith review of: Thurston's jiggling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TIINQC33}},
  note         = {Machine review of arXiv:2508.18759}
}

read the original abstract

In the 1970s Thurston introduced a technique known as ``jiggling'' which brings any triangulation into general position (a stronger version of transversality) by subdividing and perturbing. This result is now known as Thurston's jiggling lemma. In this paper we provide an alternative, more conceptual proof of the lemma. In particular we also prove the generalization to manifolds, whose proof had previously only been sketched.

Figures

Figures reproduced from arXiv: 2508.18759 by the authors.

**Figure 1.** On the left we are given the horizontal distribution ξ on R 2 and a triangulation T of R 2 . The triangulation T is not transverse to the distribution, and hence we jiggle T. We do so in two steps: we first subdivide the triangulation (middle figure), and then we perturb the image of T on the vertices (right figure). The result is a transverse triangulation. The proof starts by subdividing the triangulation in a cr… view at source ↗

**Figure 2.** 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 3.** 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_p009_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: An example of a linear 2-simplex ∆ in R 2 which is (stratified) transverse and in general position with respect to a distribution F, but whose subdivision K′ is only transverse. 5.2. Transversality of piecewise smooth maps. We now generalize all of the above notions to…

**Figure 5.** Figure 5: On the left we see a 1-simplex ∆ in R 3 that is transverse to the foliation F = ⟨∂x⟩ together with two points p and p ′ . The simplex join(p, ∆) is not transverse, whereas join(p ′ , ∆) is. On the right, in R 2 , we see that indeed π(p) ∈ π(ASpan(∆)), whereas π(p ′ ) ∈…

**Figure 6.** Figure 6: A sketch of the proof of Lemma 7.4 after projecting to R n−k . In orange we have indicated π(p) and the ε-ball around it. In green we see the point π(p ′ ) with a δ-neighborhood that does not intersect any of the ASpan(∆i) and is contained in the orange ball. Proof. Si…

**Figure 7.** Figure 7: On the left we see a subdivision K′ of the simplicial complex consisting of a single simplex ∆. The simplex ∆ is stratified transverse to the horizontal foliation F, but K′ is not. In the middle we indicate the resulting subdivision K′′ of K′ obtained by barycentricall…

**Figure 8.** Figure 8: A sketch of the proof of Theorem 8.9 when we have constructed f (i−1) and are about to construct f (i) . Below we indicate the relevant subcomplexes of K(i−1). In particular, we have indicated with green where (f (i−1), K(i−1)) is in general position. At the top we ind…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[6]

Jiggling: an h-principle without homotopical assumptions

A. Fokma, ´A. del Pino, and L. Toussaint. Jiggling: an h-principle without homotopical assump- tions. 2025. arXiv: 2501.13627 [math.GT]

work page Pith review arXiv 2025
[1]

Triangulations and the stability theorem for foliations

M.-T. Benameur. “Triangulations and the stability theorem for foliations”. In: pacific journal of mathematics 179.2 (1997), pp. 221–239

work page 1997
[2]

Bertelson and J

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

work page 2024
[3]

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
[4]

Cieliebak, Y

K. Cieliebak, Y. Eliashberg, and N. Mishachev. Introduction to the h-principle. Vol. 239. Amer- ican Mathematical Society, 2024

work page 2024
[5]

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

work page 1999
[7]

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

work page 1986
[8]

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

work page 2016

Show all 15 references

[9]

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

2009
[10]

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. 34 REFERENCES

2020
[11]

Existence of codimension-one foliations

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

1976
[12]

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
[13]

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
[14]

On C 1-complexes

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

1940
[15]

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

1957

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [6]

Jiggling: an h-principle without homotopical assumptions

A. Fokma, ´A. del Pino, and L. Toussaint. Jiggling: an h-principle without homotopical assump- tions. 2025. arXiv: 2501.13627 [math.GT]

work page Pith review arXiv 2025

[2] [1]

Triangulations and the stability theorem for foliations

M.-T. Benameur. “Triangulations and the stability theorem for foliations”. In: pacific journal of mathematics 179.2 (1997), pp. 221–239

work page 1997

[3] [2]

Bertelson and J

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

work page 2024

[4] [3]

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

[5] [4]

Cieliebak, Y

K. Cieliebak, Y. Eliashberg, and N. Mishachev. Introduction to the h-principle. Vol. 239. Amer- ican Mathematical Society, 2024

work page 2024

[6] [5]

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

work page 1999

[7] [7]

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

work page 1986

[8] [8]

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

work page 2016