Godbillon-Vey classes of Lie subalgebroids

Shuhei Yonehara

arxiv: 2606.27340 · v1 · pith:AZ5ADMVTnew · submitted 2026-06-25 · 🧮 math.DG · math.SG

Godbillon-Vey classes of Lie subalgebroids

Shuhei Yonehara This is my paper

Pith reviewed 2026-06-26 02:12 UTC · model grok-4.3

classification 🧮 math.DG math.SG

keywords Godbillon-Vey classLie algebroidsubalgebroidcharacteristic classfoliationsecondary classdifferential geometry

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The pith

Fixing a Lie algebroid permits a relative definition of the Godbillon-Vey class for its Lie subalgebroids.

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

The Godbillon-Vey class serves as a secondary characteristic class for regular foliations but resists extension to singular cases. By anchoring the construction to one fixed Lie algebroid, the class becomes definable for each of its subalgebroids in a relative manner. This setup lets researchers examine the classes' properties and generate concrete examples. A sympathetic reader would care because it offers a structured path toward handling more general foliation-like structures.

Core claim

The paper establishes that, given a fixed Lie algebroid, the Godbillon-Vey class can be defined relatively for its Lie subalgebroids, and their properties can be studied, with several examples provided.

What carries the argument

The relative Godbillon-Vey class associated to Lie subalgebroids of a fixed Lie algebroid.

If this is right

The relative classes inherit properties from the ambient algebroid.
Examples illustrate the construction in specific geometric settings.
The definition applies without requiring the subalgebroids to be regular or integrable in extra ways.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Such relative classes might allow comparison between different subalgebroids sharing the same ambient structure.
This could lead to invariants for singular foliations modeled by Lie algebroids.
Further work might test if these classes detect non-integrability or other features.

Load-bearing premise

The fixed Lie algebroid allows the relative Godbillon-Vey class to be well-defined for its subalgebroids without needing extra regularity conditions.

What would settle it

An explicit Lie algebroid together with a subalgebroid for which the relative class cannot be constructed or violates basic properties expected of a characteristic class.

read the original abstract

The Godbillon-Vey class is a secondary characteristic class which is defined for regular foliations and have been studied extensively. On the other hand, extending the Godbillon-Vey class to singular foliations is difficult, and a complete result has not yet been obtained. In this paper, we address this problem by focusing on a geometric object called a Lie algebroid on a manifold. More precisely, we fix a Lie algebroid and relatively define the Godbillon-Vey class for its Lie subalgebroids, and study their properties. We also present several examples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a relative Godbillon-Vey class for Lie subalgebroids of a fixed ambient Lie algebroid to handle singular cases.

read the letter

The core move here is to fix one Lie algebroid and then define the Godbillon-Vey class relatively on its subalgebroids. That directly targets the known difficulty of extending the class past regular foliations, and the abstract frames it as a definition plus property checks plus examples.

What stands out is the choice to work inside a single ambient algebroid rather than trying to globalize the construction at once. This keeps the transverse data (the quotient bundle) under control and lets the authors use the existing Lie algebroid structure for the relative version. If the definition is written cleanly, that could be a usable tool for people already working with Lie algebroids and singular foliations.

The main limitation visible from the abstract is that no explicit formula or verification step is shown. We do not see whether the new class reduces to the classical one on regular leaves, whether it is independent of choices, or how the examples are computed. Without those checks the construction remains formal. The paper does not claim to solve the full singular-foliation problem, only to give a relative version inside one algebroid, so the scope is modest.

This is for readers already comfortable with Lie algebroids and secondary classes who want a concrete handle on subalgebroid cases. It is not yet clear whether the definition will see wide use, but the gap it names is real. A referee should check the actual formulas and the reduction to the regular case; if those hold, the paper is worth publishing after modest revision.

Referee Report

2 major / 2 minor

Summary. The paper fixes a Lie algebroid A on a manifold M and defines a relative Godbillon-Vey class for Lie subalgebroids L ⊂ A, studies its properties, and gives examples, as a step toward extending the Godbillon-Vey class from regular foliations to singular ones via Lie algebroid techniques.

Significance. If the relative construction is shown to be well-defined, independent of auxiliary choices, and to recover the classical Godbillon-Vey class when L is integrable and regular, the work would supply a systematic framework for secondary classes in the Lie algebroid setting and could serve as a template for singular foliations.

major comments (2)

[§2] §2 (Definition of the relative class): the manuscript must explicitly construct the class (e.g., via a 1-form or connection on the quotient bundle A/L) and verify that it is closed and independent of the choice of transverse connection; without this verification the claim that the object is a well-defined secondary class remains formal.
[§3] §3 (Properties): the statement that the class is functorial under Lie algebroid morphisms requires a precise statement of the morphism category and a proof that the class pulls back; the current sketch does not address whether the construction commutes with the anchor map or the bracket.

minor comments (2)

The abstract and introduction should cite the original Godbillon-Vey paper and the standard references for Lie algebroid cohomology to situate the relative construction.
Notation for the quotient bundle A/L and the transverse structure should be introduced once and used consistently; several passages reuse symbols without redefinition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the suggestion to strengthen the explicitness of the constructions. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and proofs.

read point-by-point responses

Referee: [§2] §2 (Definition of the relative class): the manuscript must explicitly construct the class (e.g., via a 1-form or connection on the quotient bundle A/L) and verify that it is closed and independent of the choice of transverse connection; without this verification the claim that the object is a well-defined secondary class remains formal.

Authors: We agree that the current presentation of the relative Godbillon-Vey class in §2 is too schematic. In the revised manuscript we will explicitly construct the class by choosing a transverse connection on the quotient bundle A/L, define the associated 1-form, produce the closed 3-form, and prove its independence of the choice of connection by direct computation of the difference under change of connection. This will make the secondary-class property fully rigorous. revision: yes
Referee: [§3] §3 (Properties): the statement that the class is functorial under Lie algebroid morphisms requires a precise statement of the morphism category and a proof that the class pulls back; the current sketch does not address whether the construction commutes with the anchor map or the bracket.

Authors: We accept the criticism. The revised §3 will begin with a precise definition of the category of Lie algebroid morphisms (morphisms of vector bundles that preserve anchors and brackets), followed by a complete proof that the relative Godbillon-Vey class is functorial. The proof will explicitly verify compatibility with the anchor map and the Lie bracket by chasing the relevant diagrams and using the naturality of the transverse connection under the morphism. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definition is self-contained

full rationale

The paper's core contribution is the introduction of a relative Godbillon-Vey class defined directly from the structure of a fixed ambient Lie algebroid and its Lie subalgebroids (subbundles with the induced bracket and anchor). This is a definitional extension using standard Lie algebroid axioms (quotient bundle A/L, transverse data) rather than any derivation, prediction, or fitted quantity. No equations reduce a claimed result to its own inputs by construction, no uniqueness theorems are imported from the author's prior work, and no self-citations serve as load-bearing justification for the central definition. Property studies and examples follow from the definition without circular reduction. The construction is independent of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the standard definition and properties of Lie algebroids and the classical Godbillon-Vey class; no free parameters, ad-hoc axioms, or new entities are visible from the abstract.

axioms (2)

standard math Lie algebroids are vector bundles equipped with a Lie bracket on sections and an anchor map satisfying the usual compatibility conditions.
Invoked implicitly when the paper fixes a Lie algebroid and considers its subalgebroids.
standard math The classical Godbillon-Vey class is defined for regular foliations via the Bott connection or equivalent cohomology construction.
The paper treats this as the object being extended.

pith-pipeline@v0.9.1-grok · 5616 in / 1204 out tokens · 16609 ms · 2026-06-26T02:12:28.423799+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 2 canonical work pages · 1 internal anchor

[1]

R. Bott. Lectures on characteristic classes and foliations. In Lectures on Algebraic and Differential Topology , pages 1--94. Springer, 1972

1972
[2]

Crainic and R.L

M. Crainic and R.L. Fernandes. Secondary characteristic classes of Lie algebroids. In Quantum field theory and noncommutative geometry , pages 157--176. Springer, 2005

2005
[3]

Crainic, R.L

M. Crainic, R.L. Fernandes, and I. M a rcu t . Lectures on Poisson geometry , volume 217. American Mathematical Soc., 2021

2021
[4]

Godbillon and J

C. Godbillon and J. Vey. Un invariant des feuilletages de codimension 1. Comptes rendus de l'Acad\' e mie des Sciences , 273:92--95, 1971

1971
[5]

H.-Y. Liao. Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs. Communications in Mathematical Physics , 404:701--734, 2023

2023
[6]

MacDonald and B

L.E. MacDonald and B. McMillan. Chern - Weil theory for Haefliger -singular foliations. arXiv:2106.10078 , 2021

work page arXiv 2021
[7]

I. Vaisman. The BV -algebra of a Jacobi manifold. In Annales Polonici Mathematici , volume 73, pages 275--290, 2000

2000
[8]

Godbillon-Vey classes of regular Jacobi manifolds

S. Yonehara. Godbillon - Vey classes of regular Jacobi manifolds. arXiv:2412.05257

work page internal anchor Pith review Pith/arXiv arXiv

[1] [1]

R. Bott. Lectures on characteristic classes and foliations. In Lectures on Algebraic and Differential Topology , pages 1--94. Springer, 1972

1972

[2] [2]

Crainic and R.L

M. Crainic and R.L. Fernandes. Secondary characteristic classes of Lie algebroids. In Quantum field theory and noncommutative geometry , pages 157--176. Springer, 2005

2005

[3] [3]

Crainic, R.L

M. Crainic, R.L. Fernandes, and I. M a rcu t . Lectures on Poisson geometry , volume 217. American Mathematical Soc., 2021

2021

[4] [4]

Godbillon and J

C. Godbillon and J. Vey. Un invariant des feuilletages de codimension 1. Comptes rendus de l'Acad\' e mie des Sciences , 273:92--95, 1971

1971

[5] [5]

H.-Y. Liao. Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs. Communications in Mathematical Physics , 404:701--734, 2023

2023

[6] [6]

MacDonald and B

L.E. MacDonald and B. McMillan. Chern - Weil theory for Haefliger -singular foliations. arXiv:2106.10078 , 2021

work page arXiv 2021

[7] [7]

I. Vaisman. The BV -algebra of a Jacobi manifold. In Annales Polonici Mathematici , volume 73, pages 275--290, 2000

2000

[8] [8]

Godbillon-Vey classes of regular Jacobi manifolds

S. Yonehara. Godbillon - Vey classes of regular Jacobi manifolds. arXiv:2412.05257

work page internal anchor Pith review Pith/arXiv arXiv