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


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


Zbl 0398.57018
Full Text: DOI EuDML


[1] BROWN R. -Elements of Modern Topology. Mc Graw-Hill, London (1968) · Zbl 0159.52201
[2] HAEFLIGER A. -Homotopy and integrability; in Manifolds, Amsterdam 1970, Lecture Notes in Math., vol. 197, Springer-Verlag
[3] KODAIRA & SPENCER-Multifoliate Structures. Annals of Mathematics. Vol. 74, n{\(\deg\)}1, July 1961
[4] LUNDELL & WEINGRAMM.-The topology of CW-complexes. Van Nostrand Reinhold Company (1969) · Zbl 0207.21704
[5] SEGAL G. B.-Classifying spaces and spectral sequences. Publi. Math. I.H.E.S. 34, pp. 105-112 (1968) · Zbl 0199.26404
[6] SEGAL G. B. -Categories and cohomology theories. Topology 13 (1974), pp. 293-312 · Zbl 0284.55016
[7] SEGAL G. B. -Classifying spaces related to foliations. Topology vol. 17, pp. 367-382 (1978) · Zbl 0398.57018
[8] VAISMAN I. -Almost multifoliate Riemannian manifolds. Ann. Sc. Univ. ?Al. I. Cuza?, sect. I, vol. XVI (1970), pp. 97-104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.