Classifying spaces for foliations with isolated singularities

Abstract:

Let ${\Gamma ^a} \subset \Gamma$ be transitive pseudogroups on ${{\mathbf {R}}^n}$, such that, for any element $g: U \to V$ of $\Gamma$, there is a locally finite subset $S \subset U$, such that $g{|_{U - S}}$ is an element of ${\Gamma ^a}$. We construct $B\Gamma$, up to weak homotopy type, from $B{\Gamma ^a}$ and the classifying spaces of certain groups of germs. As an application, the classifying space of the pseudogroup of orientation-preserving, piecewise linear homeomorphisms between open subsets of ${\mathbf {R}}$ is weakly homotopy equivalent to $B{\mathbf {R}}{\ast }B{\mathbf {R}}$.