zbMATH — the first resource for mathematics

Rapport sur les S-atlas. (French) Zbl 0543.58003
Structure transverse des feuilletages, Toulouse 1982, Astérisque 116, 235-292 (1984).
[For the entire collection see Zbl 0534.00014.] The notion of an S-atlas generalizes the usual atlases of differentiable manifolds, and, roughly speaking, an S-atlas consists of a ”big” differentiable manifold P (which may be nonconnected and nonseparated), and a groupoid T over P (which is a generalization of the structure defined by the germs of the elements of a pseudogroup acting on P). The charts of the atlas are the connected components of P, and the elements of the groupoid are ”transition functions”. S-atlases are able to describe some nonmanifold quotients of manifolds, while still letting the differentiability properties play an important role. Particularly, this is true for transversal structures of foliations, and it is useful in order to give Lie’s third theorem for Banach-Lie groups. The main subjects developed in this paper are adapted notions of fundamental groups, and unramified and ramified covering spaces. This theory is applied to some geometric situations, and particularly, to real analytic foliations of codimension one.
Reviewer: I.Vaisman

58A05 Differentiable manifolds, foundations
58A40 Differential spaces
57R30 Foliations in differential topology; geometric theory