Brown, Ronald; Múcuk, Osman The monodromy groupoid of a Lie groupoid. (English) Zbl 0844.22006 Cah. Topologie Géom. Différ. Catég. 36, No. 4, 345-369 (1995). Given a connected Lie group \(G\), the universal cover \(\widetilde G\) of \(G\) has an essentially unique Lie group structure such that the projection \(\widetilde G \to G\) is an étale morphism, and any local morphism with domain \(G\) globalizes to a unique morphism defined on \(\widetilde G\). J. Pradines [C. R. Acad. Sci., Paris, Sér. A, 263, 907-910 (1966; Zbl 0147.41102)] announced an analogous construction for (not necessarily locally trivial) Lie groupoids \(G\) whose stars (that is, the fibres of the source projection) are connected; he called the corresponding groupoid the monodromy groupoid of \(G\) since, applied to pair groupoids \(M \times M\), for \(M\) any connected manifold, this yields a classical form of the monodromy principle.In this paper the authors give a detailed construction of the monodromy groupoid of a general Lie groupoid with connected stars, showing that the disjoint union of the universal covers of the stars may be retopologized to give a Lie groupoid structure; in view of the great variety of Lie groupoids, this is a remarkable result. They further prove a globalization result for local morphisms analogous to the standard result for local morphisms of Lie groups. The methods used apply (with suitable care and some additional technical hypotheses) equally to topological groupoids and the intermediate differentiability classes \(C^r\), \(r \geq 1\). Reviewer: K.Mackenzie (Sheffield) Cited in 4 ReviewsCited in 12 Documents MSC: 22A22 Topological groupoids (including differentiable and Lie groupoids) 58H05 Pseudogroups and differentiable groupoids Keywords:Lie group; universal cover; étale morphism; Lie groupoids; monodromy groupoid; topological groupoids Citations:Zbl 0147.41102 PDFBibTeX XMLCite \textit{R. Brown} and \textit{O. Múcuk}, Cah. Topologie Géom. Différ. Catégoriques 36, No. 4, 345--369 (1995; Zbl 0844.22006) Full Text: Numdam EuDML References: [1] Aof , M.E. - S.A.-F. Topological aspects of holonomy groupoids, PhD Thesis , University of Wales , Bangor , 1988 . MR 980945 [2] Aof , M.E.-S.A.-F. AND Brown , R. , The holonomy groupoid of a locally topological groupoid , Top. Appl. , 47 ( 1992 ) 97 - 113 . 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