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Foliations, locally Lie groupoids and holonomy. (English) Zbl 0868.22001

The Lie theory for Lie groupoids and Lie algebroids proposed by J. Pradines [C. R. Acad. Sci., Paris, Sér. A 263, 907-910 (1966; Zbl 0147.41102)] has as a consequence that a foliation of a paracompact manifold must correspond to a localized form of the Lie groupoid concept. More precisely, a locally Lie groupoid \((G,W)\) is a set \(G\) with a groupoid structure, together with a suitably symmetric subset \(W\) which generates \(G\), and a smooth structure on \(W\) with respect to which it satisfies local forms of the Lie groupoid axioms.
This paper gives a careful proof that the equivalence relation \(R_F\subseteq X\times X\) of a foliation \(F\) on a manifold \(X\) has a locally Lie groupoid structure. In conjunction with the paper [Cah. Topologie et Géom. Différ. Catégoriques 36, No. 4, 345-369 (1995; Zbl 0844.22006)] of the same authors, it gives the first full account of the holonomy and monodromy constructions of J. Pradines [op. cit.]. The paper concludes with two very clear examples of locally Lie groupoids \(R_F\) where the smooth structure does not extend to a global Lie groupoid structure on the whole of \(R_F\).

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
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References:

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