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Groupoïdes d’holonomie et classifiants. (French) Zbl 0562.57012
Structure transverse des feuilletages, Toulouse 1982, Astérisque 116, 70-97 (1984).
[For the entire collection see Zbl 0534.00014.]
This paper, a very good summary of the different types of groupoids associated with a foliation, was written by the inventor of some of these structures. This is a review article and certainly does the job. First, he explains the holonomy groupoid or graph, G of a foliation. Then, he defines the holonomy groupoid \(\Gamma_ T\) of a complete transversal T of the foliation. These groupoids are looked at in terms of different geometric structures put on the foliation. Also discussed is the problem of relating the pseudogroup of transformations of a manifold to the graph of a foliation on a compact manifold.
The classifying space BG of G is defined and invariants of the transverse structure are examined. Cohomology with \(\Gamma\)-coefficients and \(\Gamma\)-principal bundles are also looked at. A review of Milnor’s construction of classifying spaces is given in terms of holonomy groupoids. The author also interprets Steenrod’s work on bundles via groupoids. The paper concludes with a discussion of V-manifolds, their relationship with holonomy groupoids, and their classifying spaces.
Reviewer: I.S.Moskowitz

57R30 Foliations in differential topology; geometric theory
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
53C12 Foliations (differential geometric aspects)
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology