This paper is devoted to define a characteristic homomorphism for a subfoliation $(F_1,F_2)$ and to study its relation with the usual characteristic homomorphism for each foliation (as defined by Bott). Moreover, two applications are given: 1) the Yamato’s 2-codimensional foliation is shown to be no homotopic to $F_2$ in a (1,2)-codimensional subfoliation; 2) an obstruction to the existence of $d$ everywhere independent and transverse infinitesimal transformations of a foliation $F_2$ is obtained, when $F_2$ and these...

In this paper a construction of characteristic classes for a subfoliation $(F_1,F_2)$ is given by using Kamber-Tondeur’s techniques. For this purpose, the notion of $(F_1,F_2)$-foliated principal bundle, and the definition of its associated characteristic homomorphism, are introduced. The relation with the characteristic homomorphism of $F_i$-foliated bundles, $i=1,2$, the results of Kamber-Tondeur on the cohomology of $g$-$DG$-algebras. Finally, Goldman’s results on the restriction of foliated bundles to the leaves of a foliation...

For any etale topological groupoid $G$ (for example, the holonomy groupoid of a foliation), it is shown that its classifying topos is homotopy equivalent to its classifying space. As an application, we prove that the fundamental group of Haefliger for the (leaf space of) a foliation agrees with the one introduced by Van Est. We also give a new proof of Segal’s theorem on Haefliger’s classifying space $B{\Gamma }^q$.