On the de Rham theory of certain classifying spaces

In this thesis, H. Shulman proved the vanishing phenomena for characteristic classes of foliations without the usual geometric constructions of connections, curvature, etc. In this note, we present a setting for this, which was noted independently by the senior authors about two years ago. Since that time, this subject has grown considerably, so that our account here is to a large extent anachronistic. In particular, the work of Kamber and Tondeur, who combine simplicial and curvature techniques but avoid classifying spaces, as well as the work of Vey, Bott, Haefliger and others has progressed far beyond the results outlined here. Nevertheless, our ideas are on the one hand very simple, but on the other, involve technicalities that have been resolved only recently, that a short account at this time still seems worthwhile. The basic concept, which has been in the air for quite some time, and notably in the work of Deligne on mixed Hodge structures, is that the de Rham theory can be profitably employed as a tool for studying certain nonmanifolds, namely, those that are obtained as the geometric realization of a simplicial manifold.