## 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

### MSC:

 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

Zbl 0534.00014