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On the foliated products of class $$C^ 1$$. (English) Zbl 0701.57012
Let $$Diff^ 1_ c({\mathbb{R}}^ n)$$ denote the group of $$C^ 1$$- diffeomorphisms on $${\mathbb{R}}^ n$$ with compact support. It is shown that the homotopy fiber of the natural map from the classifying space of $$Diff^ 1_ c({\mathbb{R}}^ n)$$ with the discrete topology to that of $$Diff^ 1_ c({\mathbb{R}}^ n)$$ with the $$C^ 1$$-topology is acyclic. This homotopy fiber is the classifying space for the $$C^ 1$$-foliated $${\mathbb{R}}^ n$$-products with compact support. As a corollary Haefliger’s classifying space for $$\Gamma$$ $${}_ n$$-structures of class $$C^ 1$$ with trivialized normal bundles is contractible. This implies that any subbundle of the tangent bundle of a manifold is homotopic to the tangent bundle of a foliation of class $$C^ 1$$. The results should be contrasted with the fact that for higher differentiability classes this homotopy fiber has a non-trivial cohomology.
Reviewer: A.Kriegl

##### MSC:
 57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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