$b^k$-algebroids and the variety of foliation jets

Aldo Witte; \'Alvaro del Pino; Francis Bischoff

REVIEW 2 major objections 4 minor 1 cited by

$b^k$-algebroids and the variety of foliation jets

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves that b^{k+1}-type singular foliations are encoded by k-th order foliations — distributions involutive up to order k — and that, for a closed connected hypersurface, their isotopy classes form a character variety of the fund

desk verdict The b^{k+1}-type classification holds together; the only real soft spot is the unverified use of the [BBLM20] splitting theorem in the local normal form, which is fixable and not fatal. read the letter →

arxiv 2508.20241 v1 pith:6YVS3CXN submitted 2025-08-27 math.DG math.GT

Francis Bischoff , Álvaro del Pino , Aldo Witte This is my paper · ORCID

classification math.DGmath.GT MSC 53D1757R3058A20

keywords b^{k+1}-typesingularfoliationshypersurfacealgebroidsk-thorderLieRiemann-HilbertcorrespondenceAtiyahalgebroidcharactervarietyextensionclass

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 classifies singular foliations that are tangent to a submanifold W to order k — the b^{k+1}-type foliations, which include the logarithmic b-tangent structures used across geometry. The central claim is that such a foliation is fully encoded by its k-jet along W: a k-th order foliation, a distribution on the k-th order neighbourhood of W that is involutive up to order k. After choosing a tubular neighbourhood, the data is equivalently a flat connection on the bundle of k-jets of frames of the normal bundle, so the classification becomes a Riemann-Hilbert correspondence: foliations up to isomorphism are representations of the fundamental group of W in the jet group G_{k,l}, up to conjugation. For a closed connected hypersurface W, isotopy classes of the resulting b^{k+1}-type algebroids form the character variety Hom_{ν_W}(π_1(W), G_{k,1})/G^0_{k,1}, with Scott's b^{k+1}-tangent bundles as the trivial representation. The paper also resolves the extension problem: a k-th order foliation extends to order k+1 exactly when a characteristic class e(F) in H^2(W, Sym^{k+1}(ν*)⊗ν) vanishes, and this class ties together the character varieties of different orders.

What carries the argument

The load-bearing objects are: (1) the k-th order Atiyah algebroid at_k(E), the Lie algebroid of k-jets of projectable vector fields on a tubular neighbourhood E → W, whose Lie algebroid splittings are exactly the k-th order foliations; (2) the jet group G_{k,l} = J^k Diff(R^l, 0), the group of k-jets of diffeomorphisms of R^l fixing the origin, with Levy decomposition K_{k,l} ⋊ GL(R^l); (3) the k-th frame bundle Fr_k(E), principal G_{k,l}-bundle of k-jets of frames of the normal bundle, whose Atiyah algebroid is at_k(E) — so each k-th order foliation is a flat connection and holonomy is a representation of π_1(W); (4) the extension class e(F), the obstruction to extending to order k+1, given

What would settle it

On W = S^3 (trivial π_1) with trivial normal bundle, Corollary 8.42 predicts a single isotopy class of b^{k+1}-type algebroids — all isotopic to Scott's bundle. Producing two non-isotopic such algebroids would refute the classification. Or, in the genus-g surface example with trivial line bundle, the paper predicts a b^4-algebroid extends to b^5 exactly when Σ(x_i z_i − y_i w_i) = 0 (Example 7.1); computing the extension class of a parameter choice with non-zero value and finding an extension anyway would refute Corollary 4.7. A third check: Corollary 9.9 says ∫_{S^2} e(F) = 0 for every sphere

Watch

Extended reading notes

Core claim

Singular foliations of b^{k+1}-type — vector fields tangent to W to order k — correspond one-to-one with k-th order foliations (distributions involutive up to order k). With a tubular neighbourhood, these are equivalently Lie algebroid splittings of the k-th order Atiyah algebroid at_k(E) or flat connections on the k-th frame bundle Fr_k(E). Holonomy yields a Riemann-Hilbert correspondence: classes of foliations are conjugacy classes of ρ: π_1(W) → G_{k,l}; for a closed connected hypersurface, isotopy classes form Hom_{ν_W}(π_1(W), G_{k,1})/G^0_{k,1}, with Scott's bundles as the trivial class. Extension to order k+1 is obstructed by an extension class e(F) in H^2(W, Sym^{k+1}(ν*)⊗ν).

Load-bearing premise

The dictionary rests on one imported splitting theorem: locally around W, every b^{k+1}-type foliation is isomorphic to the model spanned by monomials x^I ∂_{x_i} of weight k+1 in the normal coordinates together with the tangent directions of W. If this normal form fails for some germ, the bijection with k-th order foliations collapses.

Editorial extensions

If this is right

Hypersurface algebroids of b^{k+1}-type — the generalizations of logarithmic and b-tangent bundles used in index theory, Poisson geometry, and integrable systems — are classified by the topology of W: up to isotopy they are exactly the points of Hom_{ν_W}(π_1(W), G_{k,1})/G^0_{k,1}.
Scott's b^{k+1}-tangent bundles, whose definition requires a choice of defining-function jet, all represent the same isotopy class; the auxiliary choice is immaterial.
The extension problem is decidable from a cohomology class: a k-th order foliation extends to order k+1 if and only if e(F) = 0, and when an extension exists the space of extensions is a torsor for H^1_∇(W, Sym^{k+1}(E*)⊗E) modulo the stabilizer of the holonomy.
Because e(F) is aspherical (its integral over every map of S^2 into W vanishes), the existence of higher-order extensions is governed by H^1 and H^2 of W with local coefficients — a cohomological, not homotopical, constraint.

Reading between the lines

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

Since G_{k,l} has the homotopy type of GL(R^l), the entire b^{k+1}-moduli space is assembled from cohomology classes of W with coefficients in symmetric powers of the normal bundle; the smooth structure of M near W enters only through the isomorphism class of the normal bundle and the flat connection it induces — a much coarser invariant than the full foliation data.
Lemma 10.3 implies the linear model (the trivial representation) lies in the closure of every component of the character variety; read as a deformation principle, every b^{k+1}-type algebroid degenerates to Scott's bundle, which could make the trivial representation a universal starting point for constructing new structures by unfolding along paths in the moduli space.
A testable boundary case: run the bijection for non-compact W, manifolds with corners, or normal bundles without any flat connection — cases where the imported splitting theorems' hypotheses are strained; the dictionary should break exactly where the local monomial model fails.
The extension class viewed as a section of a vector bundle over the character stack makes extendability a zero-set condition; one could seek analogous sections for higher-order deformations of other singular geometric structures built on b-type tangent bundles, transferring the paper's obstruction theory wholesale.

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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

One thing to know before you look at it: this is a serious classification paper, and the main result holds up. The authors introduce b^{k+1}-type singular foliations for a submanifold of arbitrary codimension, show they are equivalent to k-th order foliations (jets of distributions involutive up to order k), and then, after choosing a tubular neighbourhood, to Lie algebroid splittings of the k-th order Atiyah algebroid and to flat G_{k,l}-connections on the k-th order frame bundle. From there they get a Riemann-Hilbert correspondence, both topological and up to isotopy, and an extension obstruction living in H^2(W, Sym^{k+1}(ν^*)⊗ν). The classification of hypersurface algebroids in terms of a character variety is the concrete payoff, and the examples—the surface quadric, the Heisenberg obstruction, the mapping torus—are explicit and checkable. The paper is honest about the overlap with Francis and with Fischer–Laurent-Gengoux; it does not oversell its set-theoretic part.

Now the soft spot. Proposition 3.1, which gives the local normal form for Tan_k(W,F), invokes the splitting theorem [BBLM20] without checking its hypotheses. Your stress-test note is right that at points off W, the leaf L of a complementary foliation lies inside the regular leaf M\W, and the intersection A∩TL jumps from TL to 0 as you hit W. I do not think this is fatal: the transversality condition that [BBLM20] actually needs—T_xM = A_x + T_xL at all points of L—does hold, and the jump in the intersection is exactly the kind of thing that splitting theorem is designed to handle. But the text should say so. The same applies to the proof of Proposition 1.4 for hypersurface algebroids: it cites the Lie algebroid splitting theorem without stating the precise transversality condition. This is a rigor-and-readability fix, not a collapse.

If you want to be sure, check the application of the splitting theorem in the k-jet bijection of Theorem 6.19; if the local model were false, the bijection and the subsequent RH classification would fall. I don't see evidence of that. The rest of the paper—the Maurer-Cartan description, the holonomy logic via Fr_k(E), the comparison of extension classes through Van Est—is coherent and detailed. There is a lot of material here, and it is well organized.

Who is this for: anyone working on b^k geometry, singular foliations, or log-symplectic Poisson geometry. It is not light reading, but the character-variety viewpoint will probably become the standard way to think about these objects. I would send it to a serious referee—ideally someone who can check the [BBLM20] application and the Van Est comparison—rather than desk reject.

Referee Report

2 major / 4 minor

Summary. The paper introduces and classifies singular foliations of b^{k+1}-type, a class of singular foliations whose local model is the module generated by monomials of weight k+1 in the normal directions together with the tangent distribution of a submanifold W. The central result is a bijection between such foliations and k-th order foliations on the k-th order neighbourhood of W, and, after choosing a tubular neighbourhood, with Lie algebroid splittings of the k-th order Atiyah algebroid and flat G_{k,l}-connections on the k-th order frame bundle. Using topological groupoids and holonomy, the authors prove a Riemann-Hilbert correspondence, a topological refinement, and an isotopy classification. In codimension one this yields a homeomorphism between isotopy classes of hypersurface algebroids of b^{k+1}-type and a G^0_{k,1}-quotient of a representation variety of π_1(W). The paper also studies the extension problem from k-th to (k+1)-st order foliations, defining an extension class e(F) and showing it can be viewed as a section of a vector bundle over the k-jet character stack. Several explicit examples, including surface, Heisenberg, mapping-torus, and codimension-two cases, are computed.

Significance. If correct, the paper gives a substantial and coherent classification of a natural class of singular foliations and Lie algebroids, connecting them to jet geometry, Atiyah algebroids, and representation varieties of the fundamental group. The Riemann-Hilbert correspondence for k-th order foliations, the isotopy-level classification, and the extension-class theory are new and likely to be useful for later work on b^k-geometry and Poisson geometry. The paper is careful and detailed: the proofs are largely self-contained once standard splitting, Riemann-Hilbert, and Van Est inputs are accepted, and the examples in Section 7 provide concrete, checkable predictions. The explicit computation of extension classes and the description of the character stack are notable strengths.

major comments (2)

[Section 3, Proposition 3.1(a)] The proof applies the splitting theorem for singular foliations [BBLM20] to a leaf L of an auxiliary foliation C complementary to W. As written, the hypotheses of the theorem are not stated or verified. At points of L that are not on W, Tan_k(W,F) is all of TM, so L is contained in a regular leaf and is not transverse to the leaves in the usual geometric sense. This step is load-bearing: it produces the local normal form that identifies Tan_k(W,F) as b^{k+1}-type and is used in the bijection (c) and in Theorem 6.19. I suspect the intended hypothesis is the algebraic transversality condition T_xM = F_x + T_xL for all x∈L, which does hold here (trivially off W, and by complementarity of C on W). The authors should state the precise version of [BBLM20] they use and verify this condition explicitly.
[Section 8.1.6, Theorem 8.7 proof] The proof contains the assertion: 'Since V_t|_W=0, and assuming that k ≥ 1, the linear approximation of V_t vanishes, implying that ν(φ_t)=id.' This is false: in the local model, the vector field V = z ∂_y lies in Tan_k(W,F), vanishes on W, but has a nonzero first jet. The conclusion ν(φ_t)=id is nevertheless correct, and the missing justification is that the normal component of any section of Tan_k lies in I^{k+1}TM, so the induced map on the normal bundle is the identity. This step should be corrected, since Theorem 8.7 is used in the proof of Theorem 8.35 to identify the kernel of the Riemann-Hilbert fibration.

minor comments (4)

[Proposition 1.4 proof] The proof uses the notation i^!A after introducing γ^!A; the former should presumably be γ^!A. Please fix this typo.
[Section 6.2.1] 'Levy decomposition' should be 'Levi decomposition'.
[Theorem 9.3] 'cohology' should be 'cohomology'.
[Theorem 8.35 and Section 8.2.4] The notation Rep(Frk(E),g) is used, but the framing is denoted φ elsewhere; use a consistent symbol, e.g. Rep(Fr_k(E),φ).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the b^{k+1}/k-jet bijection and Riemann-Hilbert classification are derived from independent splitting and Riemann-Hilbert inputs, not from the conclusions being assumed.

full rationale

The paper's central derivation is not circular. Definition 1.5 introduces b^{k+1}-type singular foliations via an explicit local normal form; this is a definition of a class of objects, not an assumption of the classification theorem. Proposition 3.1 proves the equivalence with k-th order foliations: the forward direction forms Tan_k(W,F) and invokes the splitting theorem of [BBLM20] to obtain the local model, while the inverse direction reads off I_k(A) from the local model and checks directly that it is a k-th order foliation. The bijection I_k/P_k is then a consequence of the local model, not a restatement of it. The Riemann-Hilbert classification (Theorems 8.4, 8.35, 8.41 and Corollary 8.42) reduces, through Theorem 6.4, to the classical Riemann-Hilbert correspondence for flat principal bundles developed in Appendix C; this is an external, independently established input. The extension class e(F) (Definition 4.6, Corollary 4.7, Corollary 6.8) is the standard extension class of the Lie algebroid sequence (4.1), and its vanishing criterion is the usual splitting obstruction, not a fitted parameter or a renamed prediction. Self-citations are not load-bearing in a circular way: [BdPW23] is mentioned only as the origin of the project, and [BBLM20], despite sharing an author, is a published theorem with stated hypotheses and an independent proof, so it constitutes real external evidence under the review rules. The only notable concern is that the proof of Proposition 3.1 does not spell out the verification of the hypotheses of [BBLM20] for the auxiliary leaf L; this is a correctness or hypothesis-checking issue, not a circularity. No equation of the paper is shown to be equivalent to its own input by construction, and no fitted constant is relabelled as a prediction.

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

No free parameters are fitted to data. The listed axioms are standard mathematical inputs, none of which contain the target classification as a conclusion. The central new content is the translation between b^{k+1}-type foliations and k-jet data, plus the derived Riemann-Hilbert and extension results.

assumptions (5)

standard math Splitting theorem for Lie algebroids and singular foliations ([Duf01, Fer02, Wei00, BLM19], [BBLM20])
Used in Proposition 1.4 and Proposition 3.1 to justify the local normal forms for b^{k+1}-type algebroids and singular foliations around W.
standard math Classical Riemann-Hilbert correspondence for smooth flat connections on principal bundles (Appendix C, Theorem C.7)
Used to convert flat G_{k,l}-connections on Fr_k(E) into representations of pi_1(W), and to prove the topological and isotopy upgrade.
standard math Van Est isomorphism comparison for Lie groupoid cohomology ([WX91], [Cra03])
Used in Section 9 to relate the group cohomology class rho^* E_{k+1,l} to the de Rham extension class e(F) in H^2(W, Sym^{k+1}(nu*) tensor nu).
standard math Contractibility of the space of tubular neighbourhood embeddings and the isotopy extension theorem (Lemma B.8, [Cer61], [God07])
Needed to show that the functor on tubular neighbourhoods is well-defined and to pass from local classification to isotopy classification in Section 8.3.
domain assumption W is closed and connected, and codimension l and order k are fixed
The main classification theorems in Sections 8 and 10 assume closed and connected W; the paper explicitly restricts to connected W in Remark 1.3.

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Pith. "Pith review of $b^k$-algebroids and the variety of foliation jets." pith.science (2026). https://pith.science/paper/6YVS3CXN

@misc{pith2026250820241,
  author       = {Pith},
  title        = {Pith review of: $b^k$-algebroids and the variety of foliation jets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6YVS3CXN}},
  note         = {Machine review of arXiv:2508.20241}
}

abstract

We introduce and classify singular foliations of $b^{k+1}$-type, which formalize the properties of vector fields that are tangent to a submanifold $W \subset M$ to order $k$. When $W$ is a hypersurface, these structures are Lie algebroids generalizing the $b^{k+1}$-tangent bundles introduced by Scott. We prove that singular foliations of $b^{k+1}$-type are encoded by $k$-th order foliations: jets of distributions that are involutive up to order $k$, equivalently described as foliations on the $k$-th order neighborhood of $W$. Using this encoding, we construct topological groupoids of $k$-th order foliations and employ the holonomy invariant to show that these groupoids fiber over certain character stacks, yielding Riemann-Hilbert style classifications up to local isomorphism and isotopy. We also study the problem of extending a $k$-th order foliation to a $(k+1)$-st order foliation. We prove that this is obstructed by a characteristic class that arises as a section of a vector bundle over the relevant character stack.

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math.DG 2026-02 conditional novelty 5.0 of 10

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[1] [1]

The holonomy groupoid of a singular foliation

Iakovos Androulidakis and Georges Skandalis. The holonomy groupoid of a singular foliation. Journal für die reine und angewandte Mathematik , 2009(626):1--37, 2009

work page 2009

[2] [2]

Deformation spaces and normal forms around transversals

Francis Bischoff, Henrique Bursztyn, Hudson Lima, and Eckhard Meinrenken. Deformation spaces and normal forms around transversals. Compos. Math. , 156(4):697--732, 2020

work page 2020

[3] [3]

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Francis Bischoff, \'A lvaro del Pino, and Aldo Witte. Jets of foliations and b^k -algebroids, 2023

work page 2023

[4] [4]

Normal forms and moduli stacks for logarithmic flat connections

Francis Bischoff. Normal forms and moduli stacks for logarithmic flat connections. J. Algebraic Geom. , 34(2):331--379, 2025

work page 2025

[5] [5]

Splitting theorems for P oisson and related structures

Henrique Bursztyn, Hudson Lima, and Eckhard Meinrenken. Splitting theorems for P oisson and related structures. J. Reine Angew. Math. , 754:281--312, 2019

work page 2019

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Poisson geometry on b^k -algebroids, work in progress, 2025

Francis Bischoff and Aldo Witte. Poisson geometry on b^k -algebroids, work in progress, 2025

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Alberto Candel and Lawrence Colon. Foliations i. Graduate Studies in Mathematics , 23, 2000

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[8] [8]

Topologie de certains espaces de plongements

Jean Cerf. Topologie de certains espaces de plongements. Bulletin de la Société Mathématique de France , 89:227--380, 1961

work page 1961