[Paper Review] Holonomy and monodromy groupoids

Ronald Brown, İlhan Içen+1 more|ArXiv.org|Oct 5, 2001

Homotopy and Cohomology in Algebraic Topology16 references3 citations

TL;DR

This paper presents a universal, local-to-global construction of holonomy and monodromy groupoids for locally Lie groupoids, generalizing classical foliation invariants. By using universal properties and local admissible sections, it establishes that the monodromy groupoid admits a Lie structure via the holonomy construction, reversing the usual quotient-based approach and enabling higher-dimensional generalizations.

ABSTRACT

We outline the construction of the holonomy groupoid of a locally Lie groupoid and the monodromy groupoid of a Lie groupoid. These specialise to the well known holonomy and monodromy groupoids of a foliation, when the groupoid is just an equivalence relation.

Motivation & Objective

• To generalize the classical holonomy and monodromy groupoids of foliations to the broader setting of locally Lie groupoids.
• To establish a universal principle for extending local morphisms to global ones via the monodromy groupoid, inspired by Chevalley’s monodromy principle.
• To show that the holonomy groupoid construction is fundamental in endowing the monodromy groupoid with a Lie structure, rather than the other way around.
• To explore the potential of locally Lie groupoids as a framework for encoding local geometric structures in differential geometry and Lie theory.
• To initiate the program of higher-dimensional analogues of holonomy and monodromy using n-fold groupoids and algebraic inverses to subdivision.

Proposed method

• Define a locally Lie groupoid as a groupoid G with a subset W containing identities, equipped with a manifold structure such that the groupoid operations are as smooth as possible on W.
• Construct the holonomy groupoid Hol(G, W) as an overgroupoid of G, where W becomes an open subspace and the Lie structure extends globally via universal properties.
• Use local admissible sections—smooth sections of the source map defined on open subsets of the object space, with image in W and diffeomorphic image under the target map—to define local procedures.
• Apply left multiplication by such local sections to preserve openness and smoothness, generalizing the group action idea to groupoids.
• Define the monodromy groupoid M(G) as the universal cover of the stars of G at identities, with a universal lifting property for local morphisms.
• Prove that the holonomy construction provides the necessary topology on M(G) to make it a Lie groupoid, even in the non-trivial holonomy case.

Experimental results

Research questions

• RQ1 How can the holonomy and monodromy groupoids of a foliation be generalized beyond equivalence relations to arbitrary locally Lie groupoids?
• RQ2 What is the role of local admissible sections in constructing the Lie structure on the monodromy groupoid?
• RQ3 Can the monodromy groupoid be universally characterized as the domain of extension for local morphisms, and how does this relate to classical monodromy principles?
• RQ4 What is the relationship between locally Lie groupoids and Lie algebroids, and how useful is this notion for encoding local geometric data?
• RQ5 What are the prospects for higher-dimensional analogues of holonomy and monodromy using n-fold groupoids and algebraic inverses to subdivision?

Key findings

• The monodromy groupoid M(G) of a Lie groupoid G can be endowed with a Lie groupoid structure via the holonomy construction, even when holonomy is non-trivial.
• The holonomy groupoid Hol(G, W) is constructed as an overgroupoid of G, with W as an open subspace, and serves as the universal domain for extending local morphisms.
• The construction of the Lie structure on M(G) relies crucially on the holonomy groupoid, reversing the standard quotient-based approach where holonomy is derived from monodromy.
• Local admissible sections are essential for preserving smoothness and openness in groupoid operations, generalizing the role of left multiplication in topological groups.
• The universal lifting property of M(G) ensures that any local morphism defined on G extends uniquely to a global morphism on M(G), realizing a generalized monodromy principle.
• The framework suggests a natural path toward higher-dimensional analogues using n-fold groupoids and algebraic inverses to subdivision, potentially linking to non-abelian homotopy theory and curvature in principal bundles.