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.

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 |