## Modèle de Segal pour les structures multifeuilletées. (Segal model for multifoliated structures).(French)Zbl 0603.57018

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 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.
Reviewer: S.Jekel

### MSC:

 57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology

Zbl 0398.57018
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### References:

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