On the foliated products of class \(C^ 1\) (Q914198)

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.