On the homotopy and cohomology of the classifying space of Riemannian foliations

Abstract:

Let $G$ be a closed subgroup of the general linear group. Let $B\tilde \Gamma _G^q$ be the classifying space for $G$-foliated microbundles of rank $q$. (The $G$-foliation is not assumed to be integrable.) The homotopy fiber $F\tilde \Gamma _G^q$ of the classifying map $\nu :B\tilde \Gamma _G^q \to BG$ is shown to be $(q - 1)$-connected. For the orthogonal group, this implies $FR{\Gamma ^q}$ is $(q - 1)$-connected. The indecomposable classes in ${H^ * }(R{W_q})$ therefore are mapped to linearly independent classes in ${H^ * }(FR{\Gamma ^q})$; the indecomposable variable classes are mapped to independently variable classes. Related results on the homotopy groups ${\pi _ * }(FR{\Gamma ^q})$ also follow.