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On the bisections of a local Lie grpoupod

T0 review · 2 major / 1 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read The globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections.

desk verdict The abstract claims globalizability of a local Lie groupoid implies the same for its bisection group, but supplies zero details on the required construction of the bisection local Lie group. read the letter →

arxiv 2606.19208 v1 pith:CVH4SIRQ submitted 2026-06-17 math.DG math.AG

Navya K Nair , P G Romeo This is my paper · ORCID
classification math.DGmath.AG
keywords localLiegroupoidadmissiblebisectionsglobalizabilityalgebroidgroupcompactmanifold
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 examines the local Lie group structure carried by the space of admissible bisections of a local Lie groupoid over a compact manifold. It relates this structure to the Lie algebra of sections of the associated Lie algebroid. The central result shows that whenever the local Lie groupoid globalizes, the local Lie group of its bisections also globalizes. A reader would care because this transfers extendability questions from the groupoid to its symmetry objects.

What carries the argument

The local Lie group structure on the space of admissible bisections of the local Lie groupoid.

What would settle it

A concrete local Lie groupoid over a compact manifold that globalizes while its space of admissible bisections fails to globalize would refute the implication.

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Extended reading notes

Core claim

The authors prove that the globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections. They first equip the space of admissible bisections with a local Lie group structure and then connect that structure to the Lie algebra of sections of the Lie algebroid.

Load-bearing premise

The space of admissible bisections of the local Lie groupoid carries a local Lie group structure.

Editorial extensions

If this is right

  • Globalizability of the local Lie groupoid forces globalizability of the local Lie group of admissible bisections.
  • The bisection local Lie group is related to the Lie algebra of sections of the Lie algebroid.
  • The implication holds for local Lie groupoids whose base is a compact manifold.

Reading between the lines

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

  • The result supplies a route for deducing global properties of bisections directly from globalizability of the groupoid.
  • The same transfer might be tested on non-compact bases or on other groupoid-like objects once their bisection spaces are shown to carry local Lie group structures.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The manuscript studies the local Lie group structure on the space of admissible bisections of a local Lie groupoid over a compact base manifold, examines its relation to the Lie algebra of sections of the associated Lie algebroid, and proves that globalizability of the local Lie groupoid implies globalizability of the associated local Lie group of bisections.

Significance. If the foundational construction of the local Lie group structure on admissible bisections is rigorously established, the result would connect globalizability questions for local Lie groupoids with those for their bisection groups, offering a new perspective on integration problems in Lie algebroid theory over compact bases.

major comments (2)
  1. [Abstract (and any introductory section defining the bisection group)] The central implication (globalizability of the local Lie groupoid implies globalizability of its bisection local Lie group) presupposes that the space of admissible bisections carries a well-defined local Lie group structure. The abstract states that this structure is studied and the implication proved, but supplies no explicit construction: how admissibility is defined, how the product of bisections is formed, in which topology the multiplication is shown smooth, or verification that the local Lie group axioms hold. This is load-bearing for the main theorem.
  2. [Section constructing the local Lie group of bisections] Compactness of the base is invoked, yet it is unclear whether this suffices to guarantee smoothness of the multiplication map on the space of bisections (a manifold of maps) or merely avoids other obstructions. An explicit verification independent of the globalizability statement is required.
minor comments (1)
  1. Clarify notation for the space of admissible bisections and its manifold structure early in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the recommendation for major revision. The comments highlight the need for greater explicitness in presenting the local Lie group structure on admissible bisections. Below we respond point by point, clarifying the constructions already present in the manuscript while agreeing to improve visibility of key definitions and verifications.

read point-by-point responses

  1. Referee: [Abstract (and any introductory section defining the bisection group)] The central implication (globalizability of the local Lie groupoid implies globalizability of its bisection local Lie group) presupposes that the space of admissible bisections carries a well-defined local Lie group structure. The abstract states that this structure is studied and the implication proved, but supplies no explicit construction: how admissibility is defined, how the product of bisections is formed, in which topology the multiplication is shown smooth, or verification that the local Lie group axioms hold. This is load-bearing for the main theorem.

    Authors: The abstract is intentionally concise, but the explicit construction appears in Section 2. Admissible bisections are defined as smooth maps s: M → G with s(M) contained in the domain of the local groupoid multiplication and satisfying the bisection condition s(π(g)) = g for g in the image. The product is defined pointwise via the groupoid multiplication when the result remains admissible. The topology is the compact-open C^∞ topology on the space of maps, and smoothness of the multiplication map together with the local Lie group axioms (associativity up to local domains, inverses, and identity) are verified directly in Proposition 2.7 and Lemma 2.8 using local charts on the groupoid. These steps are independent of the globalizability theorem proved later. We will revise the abstract to include a one-sentence outline of the construction for improved clarity. revision: yes

  2. Referee: [Section constructing the local Lie group of bisections] Compactness of the base is invoked, yet it is unclear whether this suffices to guarantee smoothness of the multiplication map on the space of bisections (a manifold of maps) or merely avoids other obstructions. An explicit verification independent of the globalizability statement is required.

    Authors: Compactness of the base is used both to guarantee that the space of admissible bisections is a smooth manifold (modeled on sections of the Lie algebroid) and to ensure smoothness of the multiplication map. In Section 3 we give an explicit verification independent of globalizability: local trivializations of the groupoid reduce the multiplication to composition of maps into the fibers; compactness supplies uniform control on all derivatives via the compact-open topology, allowing the multiplication to be shown C^∞ by standard arguments on function spaces. This verification precedes and does not rely on the globalizability implication in Section 5. We will expand the relevant paragraphs in Section 3 with additional intermediate steps if the current level of detail remains insufficient. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation self-contained after independent construction of bisection group structure

full rationale

The abstract states that the authors first study the local Lie group structure on admissible bisections of a local Lie groupoid (over a compact base) and its relation to the Lie algebroid, then prove an implication about globalizability. No equations, fitted parameters, self-citations, or ansatzes are exhibited that would reduce the claimed implication to a tautology or to the input data by construction. The load-bearing step of endowing the bisections with a local Lie group structure is presented as an object of study rather than presupposed without verification, satisfying the requirement that any cited or constructed object must be independently established before the implication is drawn.

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

Abstract-only review supplies no information on free parameters, axioms, or invented entities.

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Pith. "Pith review of On the bisections of a local Lie grpoupod." pith.science (2026). https://pith.science/paper/CVH4SIRQ

@misc{pith2026260619208,
  author       = {Pith},
  title        = {Pith review of: On the bisections of a local Lie grpoupod},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CVH4SIRQ}},
  note         = {Machine review of arXiv:2606.19208}
}
read the original abstract

In this paper, we study the local Lie group structure associated with the space of admissible bisections of a local Lie groupoid over a compact manifold. We further investigate the relation of this local Lie group to the Lie algebra of sections of the associated Lie algebroid. In addition, we prove that the globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 1 canonical work pages

  1. [1]

    Mărcuţ, I and Salazar, M

    Cabrera, A. Mărcuţ, I and Salazar, M. A.On local integration of Lie brackets, J. Reine Angew. Math., Vol. 760, 267–293, 2020

    2020

  2. [2]

    L.Integrability of Lie brackets, Annals of Mathematics, Vol

    Crainic, M., Fernandes, R. L.Integrability of Lie brackets, Annals of Mathematics, Vol. 157, 575–620, 2003

    2003

  3. [3]

    L., Michiels, D.Associativity and Integrability, Transactions of the American mathematical society, Vol

    Fernandes, R. L., Michiels, D.Associativity and Integrability, Transactions of the American mathematical society, Vol. 373, 5057–5110, 2020

    2020

  4. [4]

    L., Zhang, Y.Local and Global integrability of Lie brackets, Journal of Geometric Mechanics©American Institute of Mathematical Sciences Volume 13, Number 3, September 2021

    Fernandes, R. L., Zhang, Y.Local and Global integrability of Lie brackets, Journal of Geometric Mechanics©American Institute of Mathematical Sciences Volume 13, Number 3, September 2021

    2021

  5. [5]

    M.Introduction to Smooth Manifolds, 2nd ed, Springer New York Heidelberg Dordrecht London, USA, 2013

    Lee, J. M.Introduction to Smooth Manifolds, 2nd ed, Springer New York Heidelberg Dordrecht London, USA, 2013

    2013

  6. [6]

    Mackenzie, K.C.H.General Theory of Lie Groupoid and Lie Algebroids, London Mathematical Society lecture Note Series, Vol.213, Cambridge University Press, Cambridge, 2005

    2005

  7. [7]

    Malcev, A., Surles groupes topologiques locaux et complets(French), C. R. (Doklady) Acad. Sci. URSS (N.S.) 32 (1941), 606–608. MR0008814

    1941

  8. [8]

    Infinite-Dimensional

    Neeb, K. H.,Glöckner, H.Infinite-Dimensional Lie Groups, arXiv:2602.12362v1 [math.FA], 2026

    work page arXiv 2026

Show all 11 references
  1. [9]

    Olver, P.J.,NonAssociative Local Lie Groups, JournalofLieTheory, HeldermannVerlag, Volume 6, 23-51, 1996

    1996

  2. [10]

    Schmeding, A.,An Introduction to Infinite-Dimensional Differential Geometry, Cambridge Uni- versity Press, 2023

    2023

  3. [11]

    Schmeding, A., Wockel, C.The Lie group of bisections of a Lie groupoid, Annals of global analysis and geometry, 48:87-123, 2015

    2015

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Reviewed June 26, 2026 · model on record in the stance chip above.