Modèle de Segal pour les structures multifeuilletées. (Segal model for multifoliated structures) (Q1082657)

Let \(\Gamma \subset \Gamma_ n\) be a subgroupoid containing the diagonal of the topological groupoid of local smooth diffeomorphisms of \({\mathbb{R}}^ n\). Then \(B\Gamma_ n\) is Haefliger's classifying space for codimension-n smooth foliations. The main result of this paper is that \(B\Gamma_ n\) is weakly homotopy equivalent to the discrete monoid of self-embeddings of \({\mathbb{R}}^ n\) all of whose germs are in \(\Gamma\). This generalizes a result of \textit{G. Segal} [Topology 17, 367-382 (1978; Zbl 0398.57018)]. The generalization comes out of carefully investigating and working out details of Segal's main lemma: an almost locally trivial map with contractible fibres is a weak homotopy equivalence. The author relates his result to the classification of multifoliations.