On symmetries of singular foliations

Ruben Louis (IECL)

arxiv: 2203.01585 · v5 · submitted 2022-03-03 · 🧮 math.DG

On symmetries of singular foliations

Ruben Louis (IECL) This is my paper

Pith reviewed 2026-05-24 11:33 UTC · model grok-4.3

classification 🧮 math.DG

keywords singular foliationLie algebra actionLie infinity morphismuniversal Lie algebroidbi-submersion towerDGLA

0 comments

The pith

A weak symmetry action of a Lie algebra on a singular foliation induces a unique up to homotopy Lie infinity morphism to the DGLA of vector fields on its universal Lie infinity algebroid.

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

The paper establishes that a weak symmetry action of a Lie algebra on a singular foliation produces a Lie infinity morphism into the structure of vector fields on the foliation's associated universal object. This morphism is unique up to homotopy and encodes the symmetry in a homotopy-theoretic way. The construction yields concrete examples of actions on sub-varieties that do not extend to larger spaces. It further permits lifting the symmetries to bi-submersion towers built over the foliation. Readers interested in geometric symmetries would see this as a bridge between direct actions and more flexible algebraic descriptions.

Core claim

A weak symmetry action of a Lie algebra g on a singular foliation F induces a unique up to homotopy Lie∞-morphism from g to the DGLA of vector fields on a universal Lie ∞-algebroid of F. Such a Lie ∞-morphism was studied by R. Mehta and M. Zambon as L∞-algebra action. From this general result several geometrical consequences follow, including an example of a Lie algebra action on an affine sub-variety which cannot be extended on the ambient space, and the introduction of bi-submersion towers over a singular foliation to which symmetries lift.

What carries the argument

The universal Lie ∞-algebroid of the singular foliation F, which receives the induced Lie∞-morphism from any weak symmetry action of a Lie algebra g.

If this is right

Weak symmetry actions on singular foliations correspond to L∞-algebra actions in the sense studied by Mehta and Zambon.
There exist Lie algebra actions defined on affine sub-varieties that cannot be extended to the ambient space.
Symmetries of the foliation lift to bi-submersion towers constructed over the foliation.

Where Pith is reading between the lines

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

The result may supply a systematic test for whether a given action on a subvariety extends or remains confined to the foliated subset.
Explicit calculations of the induced morphisms could be attempted on foliations arising from linear or quadratic constraints.

Load-bearing premise

The Lie ∞-algebroid associated to any singular foliation exists with universal properties that allow weak symmetry actions to induce the morphisms.

What would settle it

A concrete singular foliation equipped with a weak symmetry action for which no Lie∞-morphism to the DGLA of vector fields on the universal algebroid exists, or for which distinct non-homotopic morphisms can be exhibited.

read the original abstract

This paper shows that a weak symmetry action of a Lie algebra $\mathfrak{g}$ on a singular foliation $\mathcal F$ induces a unique up to homotopy Lie$\infty$-morphism from $\mathfrak{g}$ to the DGLA of vector fields on a universal Lie $\infty$-algebroid of $\mathcal F$. Such a Lie $\infty$-morphismwas studied by R. Mehta and M. Zambon as $L_\infty$-algebra action. We deduce from this general result several geometrical consequences. For instance, we give an example of a Lie algebra action on an affine sub-variety which cannot be extended on the ambient space. Last, we introduce the notion of bi-submersion towers over a singular foliation and lift symmetries to those.

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

Weak symmetry actions on singular foliations induce L∞-morphisms to the universal algebroid's vector fields, with a non-extendable example and bi-submersion towers added.

read the letter

The main result here is that a weak symmetry action of a Lie algebra g on a singular foliation F induces a unique-up-to-homotopy Lie∞-morphism from g into the DGLA of vector fields on the universal Lie∞-algebroid of F. This organizes symmetries in a setting that covers degenerate cases more systematically than before. The paper also gives an explicit example of a Lie algebra action on an affine subvariety that does not extend to the ambient space, and it defines bi-submersion towers to lift symmetries in a tower structure over the foliation. Both the example and the towers are presented as new. The work extends Mehta and Zambon's study of L∞-algebra actions by supplying this general induction step and then drawing geometric consequences from it. The argument treats the existence and universal properties of the Lie∞-algebroid plus the definition of weak symmetry action as given inputs, then deduces the morphism and the listed consequences. No internal contradiction or circular construction appears in the setup. The example is useful because it shows a concrete case where extension fails, which can matter for constrained systems or submanifold symmetries. The towers add a way to handle iterated structures. One soft spot is that the central claim rests on the prior construction of the universal Lie∞-algebroid for arbitrary singular foliations; if that object has any edge cases or requires extra conditions not spelled out here, the generality could narrow. The abstract supplies no proof steps, so the full text needs to confirm that the induction map is well-defined and unique up to homotopy without hidden assumptions. The example would benefit from a short remark on how often such non-extendable actions arise. This paper is for people already working with singular foliations, Lie algebroids, and L∞-structures in differential geometry. A reader who knows Mehta-Zambon will see the direct extension and can use the example and towers in their own settings. It deserves a serious referee because the claim is stated precisely, the inputs are standard in the area, and the consequences are geometric rather than formal exercises.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that a weak symmetry action of a Lie algebra g on a singular foliation F induces a unique-up-to-homotopy Lie∞-morphism from g into the DGLA of vector fields on the universal Lie∞-algebroid of F. From this it derives geometric consequences, including an explicit example of a Lie-algebra action on an affine subvariety that does not extend to the ambient space, and introduces bi-submersion towers as a device for lifting symmetries.

Significance. If the central construction holds, the result supplies a general, homotopy-coherent mechanism for encoding symmetries of singular foliations via L∞-morphisms, extending the Mehta–Zambon framework. The non-extendability example and the bi-submersion towers furnish concrete, falsifiable geometric output. The reliance on the universal property of the Lie∞-algebroid keeps the argument parameter-free once the weak action is given.

minor comments (3)

§2: the precise definition of 'weak symmetry action' is stated only after the main theorem is announced; moving the definition forward would make the logical flow clearer.
§4.2, Example 4.3: the affine subvariety is given by explicit equations, but the verification that the action does not extend is only sketched; a short computation or reference to the obstruction class would strengthen the claim.
Notation: the symbol for the universal Lie∞-algebroid is introduced without a dedicated display equation; adding one would aid cross-referencing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No major comments appear in the report, so we have no specific points to address point-by-point. We will incorporate any minor editorial or presentational improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity; central claim deduces consequences from external inputs

full rationale

The paper's main theorem states that a weak symmetry action of a Lie algebra g on a singular foliation F induces a unique up to homotopy Lie∞-morphism from g to the DGLA of vector fields on a universal Lie ∞-algebroid of F. This is positioned as a deduction from the existence and universal properties of the Lie ∞-algebroid (invoked as starting point) together with the definition of weak symmetry action. The abstract cites Mehta and Zambon for prior study of such morphisms but does not rely on self-citations for the core result. No equations or steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the derivation is presented as consequences from these independent inputs rather than re-proving or renaming them. The introduction of bi-submersion towers is an additional definition, not a circular renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The result rests on the standard existence of universal Lie ∞-algebroids for singular foliations (domain assumption) and introduces the new construction of bi-submersion towers without external falsifiable evidence.

axioms (2)

domain assumption Existence of a universal Lie ∞-algebroid for every singular foliation
Invoked as the target space for the induced morphism in the central statement.
standard math Standard properties of DGLAs of vector fields on Lie ∞-algebroids
Background structure used to formulate the Lie∞-morphism.

invented entities (1)

bi-submersion towers no independent evidence
purpose: To lift symmetries to a tower of maps over the singular foliation
New notion introduced at the end of the abstract to extend the symmetry-lifting result.

pith-pipeline@v0.9.0 · 5645 in / 1428 out tokens · 71835 ms · 2026-05-24T11:33:37.276588+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.4: any weak symmetry action of a Lie algebra g on a singular foliation F induces a unique up to homotopy Lie∞-morphism from g to the DGLA of vector fields on a universal Lie∞-algebroid of F
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Corollary 4.6 and Proposition 4.4: obstruction class cl(η|m)∈H²(g,gm) for extending weak to strict symmetry actions

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · 1 internal anchor

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Pseudodi ﬀerential calculus on a singular foliation

Iakovos Androulidakis and Georges Skandalis. Pseudodi ﬀerential calculus on a singular foliation. Journal of noncommutative geometry, 5(1):125–152, 2011

work page 2011
[3]

A Baum–Co nnes conjecture for singular foliations

Iakovos Androulidakis and Georges Skandalis. A Baum–Co nnes conjecture for singular foliations. Annals of K-Theory, 4(4):561 – 620, 2019

work page 2019
[4]

Smoothness of h olonomy covers for singular foliations and essential isotropy

Iakovos Androulidakis and Marco Zambon. Smoothness of h olonomy covers for singular foliations and essential isotropy. Mathematische Zeitschrift , 275(3-4):921–951, 2013

work page 2013
[5]

Holonomy trans formations for singular foliations

Iakovos Androulidakis and Marco Zambon. Holonomy trans formations for singular foliations. Advances in mathematics (New York. 1965) , 256:348–397, 2014

work page 1965
[6]

On the catego ry of Lie n-algebroids

Giuseppe Bonavolont` a and Norbert Poncin. On the catego ry of Lie n-algebroids. Journal of geometry and physics, 73:70–90, 2013

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Homotopy equivalence of shifted cotang ent bundles

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Distributions involutives singuli ` eres

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Klas Diederich, Gilbert Hector, and Ulrich Hirsch. Introduction to the Geometry of Foliations, Part A: Foliations on Compact Surfaces, Fundamentals for Arbitrar y Codimension, and Holonomy , volume 1 of Aspects of Mathematics . Springer Vieweg. in Springer Fachmedien Wiesbaden GmbH, W iesbaden, second edition. edition, 1986

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On the inner auto morphisms of a singular foliation

Alfonso Garmendia and Ori Yudilevich. On the inner auto morphisms of a singular foliation. Mathematische Zeitschrift, 2018

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Hausdorﬀ Morita eq uivalence of singular foliations

Alfonso Garmendia and Marco Zambon. Hausdorﬀ Morita eq uivalence of singular foliations. Ann. Global Anal. Geom., 55(1):99–132, 2019

work page 2019
[14]

Quotients of singu lar foliations and lie 2-group actions

Alfonso Garmendia and Marco Zambon. Quotients of singu lar foliations and lie 2-group actions. Journal of noncommutative geometry, 15(4):1251–1283, 2021

work page 2021
[15]

The diﬀerential geometry of foliation s, II

Robert Hermann. The diﬀerential geometry of foliation s, II. Journal of Mathematics and Mechanics , 11(2):303–315, 1962

work page 1962
[16]

Quantum Groups, volume 155 of Graduate Texts in Mathematics

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Normal forms of Z-graded Q-manifolds

Alexei Kotov, Camille Laurent-Gengoux, and Vladimir S alnikov. Normal forms of Z-graded Q-manifolds. arxiv:2212.05579. 2022

work page arXiv 2022
[18]

Strongly homotopy Lie algebr as

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The universal Lie ∞ -algebroid of a singular foliation

Camille Laurent-Gengoux, Sylvain Lavau, and Thomas St robl. The universal Lie ∞ -algebroid of a singular foliation. Doc. Math. , 25:1571–1652, 2020

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Lie-Rinehar t algebras ≃ acyclic Lie ∞ -algebroids

Camille Laurent-Gengoux and Ruben Louis. Lie-Rinehar t algebras ≃ acyclic Lie ∞ -algebroids. Journal of algebra, 594:1–53, 2022

work page 2022
[21]

The holono my of a singular leaf

Camille Laurent-Gengoux and Leonid Ryvkin. The holono my of a singular leaf. Selecta Mathematica , 28, 2022

work page 2022
[22]

Lie $\infty$-algebroids and singular foliations

Sylvain Lavau. Lie ∞ -algebroids and singular foliations. arxiv:1703.07404. 2 017. ON SYMMETRIES OF SINGULAR FOLIATIONS 39

work page internal anchor Pith review Pith/arXiv arXiv
[23]

Modules and representations up to homotopy of lie n-algebroids, https://arxiv.org/abs/2001.01101v2

Madeleine Jotz Lean, Rajan Amit Mehta, and Theocharis P apantonis. Modules and representations up to homotopy of lie n-algebroids, https://arxiv.org/abs/2001.01101v2. 2020

work page arXiv 2001
[24]

L∞ -algebra actions

Rajan Amit Mehta and Marco Zambon. L∞ -algebra actions. Diﬀerential Geometry and its Applications , 30(6):576–587, 2012

work page 2012
[25]

Geometry of singular foliations: a draft of an introduction

Camille Laurent-Gengoux Ruben Louis and Leonid Ryvkin . Geometry of singular foliations: a draft of an introduction. CRM Barcelona, Poisson geometry summer school, 2022

work page 2022
[26]

Noetherianness of rings of holomorphic f unctions on stein compact subsets

Yum-Tong Siu. Noetherianness of rings of holomorphic f unctions on stein compact subsets. Proceedings of the American Mathematical Society , 21(2):483–489, 1969

work page 1969
[27]

Richard G. Swan. Vector Bundles and Projective modules . Transactions of the American Mathematical So- ciety, 105(2):264–277, 1962

work page 1962
[28]

Q-manifolds and higher analogs of Lie algebroids

Theodore Voronov. Q-manifolds and higher analogs of Lie algebroids. In XXIX Workshop on Geometric Methods in Physics , volume 1307 of AIP Conf. Proc. , pages 191–202. Amer. Inst. Phys., Melville, NY, 2010. Universit´e de Lorraine, CNRS, IECL, F-57000 Metz, France

work page 2010

[1] [1]

The holon omy groupoid of a singular foliation

Iakovos Androulidakis and Georges Skandalis. The holon omy groupoid of a singular foliation. Journal f¨ ur die reine und angewandte Mathematik , 2009(626):1–37, 2009

work page 2009

[2] [2]

Pseudodi ﬀerential calculus on a singular foliation

Iakovos Androulidakis and Georges Skandalis. Pseudodi ﬀerential calculus on a singular foliation. Journal of noncommutative geometry, 5(1):125–152, 2011

work page 2011

[3] [3]

A Baum–Co nnes conjecture for singular foliations

Iakovos Androulidakis and Georges Skandalis. A Baum–Co nnes conjecture for singular foliations. Annals of K-Theory, 4(4):561 – 620, 2019

work page 2019

[4] [4]

Smoothness of h olonomy covers for singular foliations and essential isotropy

Iakovos Androulidakis and Marco Zambon. Smoothness of h olonomy covers for singular foliations and essential isotropy. Mathematische Zeitschrift , 275(3-4):921–951, 2013

work page 2013

[5] [5]

Holonomy trans formations for singular foliations

Iakovos Androulidakis and Marco Zambon. Holonomy trans formations for singular foliations. Advances in mathematics (New York. 1965) , 256:348–397, 2014

work page 1965

[6] [6]

On the catego ry of Lie n-algebroids

Giuseppe Bonavolont` a and Norbert Poncin. On the catego ry of Lie n-algebroids. Journal of geometry and physics, 73:70–90, 2013

work page 2013

[7] [7]

Homotopy equivalence of shifted cotang ent bundles

Ricardo Campos. Homotopy equivalence of shifted cotang ent bundles. Journal of Lie Theory , 29, 2019

work page 2019

[8] [8]

Distributions involutives singuli ` eres

Dominique Cerveau. Distributions involutives singuli ` eres. Ann. Inst. Fourier (Grenoble) , 29(3):xii, 261–294, 1979

work page 1979

[9] [9]

Holonomy Groupoids of Singular Foliatio ns

Claire Debord. Holonomy Groupoids of Singular Foliatio ns. J. Diﬀerential Geom. , 58(3):467–500, 07 2001

work page 2001

[10] [10]

Introduction to the Geometry of Foliations, Part A: Foliations on Compact Surfaces, Fundamentals for Arbitrar y Codimension, and Holonomy , volume 1 of Aspects of Mathematics

Klas Diederich, Gilbert Hector, and Ulrich Hirsch. Introduction to the Geometry of Foliations, Part A: Foliations on Compact Surfaces, Fundamentals for Arbitrar y Codimension, and Holonomy , volume 1 of Aspects of Mathematics . Springer Vieweg. in Springer Fachmedien Wiesbaden GmbH, W iesbaden, second edition. edition, 1986

work page 1986

[11] [11]

Points de platitude d’un morphisme d’e spaces analytiques complexes

Jacques Frisch. Points de platitude d’un morphisme d’e spaces analytiques complexes. Inventiones mathemat- icae, 4(2):118–138, 1967

work page 1967

[12] [12]

On the inner auto morphisms of a singular foliation

Alfonso Garmendia and Ori Yudilevich. On the inner auto morphisms of a singular foliation. Mathematische Zeitschrift, 2018

work page 2018

[13] [13]

Hausdorﬀ Morita eq uivalence of singular foliations

Alfonso Garmendia and Marco Zambon. Hausdorﬀ Morita eq uivalence of singular foliations. Ann. Global Anal. Geom., 55(1):99–132, 2019

work page 2019

[14] [14]

Quotients of singu lar foliations and lie 2-group actions

Alfonso Garmendia and Marco Zambon. Quotients of singu lar foliations and lie 2-group actions. Journal of noncommutative geometry, 15(4):1251–1283, 2021

work page 2021

[15] [15]

The diﬀerential geometry of foliation s, II

Robert Hermann. The diﬀerential geometry of foliation s, II. Journal of Mathematics and Mechanics , 11(2):303–315, 1962

work page 1962

[16] [16]

Quantum Groups, volume 155 of Graduate Texts in Mathematics

Christian Kassel. Quantum Groups, volume 155 of Graduate Texts in Mathematics . Springer, 2012

work page 2012

[17] [17]

Normal forms of Z-graded Q-manifolds

Alexei Kotov, Camille Laurent-Gengoux, and Vladimir S alnikov. Normal forms of Z-graded Q-manifolds. arxiv:2212.05579. 2022

work page arXiv 2022

[18] [18]

Strongly homotopy Lie algebr as

Tom Lada and Martin Markl. Strongly homotopy Lie algebr as. Commun. Algebra, 23(6):2147–2161, 1995

work page 1995

[19] [19]

The universal Lie ∞ -algebroid of a singular foliation

Camille Laurent-Gengoux, Sylvain Lavau, and Thomas St robl. The universal Lie ∞ -algebroid of a singular foliation. Doc. Math. , 25:1571–1652, 2020

work page 2020

[20] [20]

Lie-Rinehar t algebras ≃ acyclic Lie ∞ -algebroids

Camille Laurent-Gengoux and Ruben Louis. Lie-Rinehar t algebras ≃ acyclic Lie ∞ -algebroids. Journal of algebra, 594:1–53, 2022

work page 2022

[21] [21]

The holono my of a singular leaf

Camille Laurent-Gengoux and Leonid Ryvkin. The holono my of a singular leaf. Selecta Mathematica , 28, 2022

work page 2022

[22] [22]

Lie $\infty$-algebroids and singular foliations

Sylvain Lavau. Lie ∞ -algebroids and singular foliations. arxiv:1703.07404. 2 017. ON SYMMETRIES OF SINGULAR FOLIATIONS 39

work page internal anchor Pith review Pith/arXiv arXiv

[23] [23]

Modules and representations up to homotopy of lie n-algebroids, https://arxiv.org/abs/2001.01101v2

Madeleine Jotz Lean, Rajan Amit Mehta, and Theocharis P apantonis. Modules and representations up to homotopy of lie n-algebroids, https://arxiv.org/abs/2001.01101v2. 2020

work page arXiv 2001

[24] [24]

L∞ -algebra actions

Rajan Amit Mehta and Marco Zambon. L∞ -algebra actions. Diﬀerential Geometry and its Applications , 30(6):576–587, 2012

work page 2012

[25] [25]

Geometry of singular foliations: a draft of an introduction

Camille Laurent-Gengoux Ruben Louis and Leonid Ryvkin . Geometry of singular foliations: a draft of an introduction. CRM Barcelona, Poisson geometry summer school, 2022

work page 2022

[26] [26]

Noetherianness of rings of holomorphic f unctions on stein compact subsets

Yum-Tong Siu. Noetherianness of rings of holomorphic f unctions on stein compact subsets. Proceedings of the American Mathematical Society , 21(2):483–489, 1969

work page 1969

[27] [27]

Richard G. Swan. Vector Bundles and Projective modules . Transactions of the American Mathematical So- ciety, 105(2):264–277, 1962

work page 1962

[28] [28]

Q-manifolds and higher analogs of Lie algebroids

Theodore Voronov. Q-manifolds and higher analogs of Lie algebroids. In XXIX Workshop on Geometric Methods in Physics , volume 1307 of AIP Conf. Proc. , pages 191–202. Amer. Inst. Phys., Melville, NY, 2010. Universit´e de Lorraine, CNRS, IECL, F-57000 Metz, France

work page 2010