A characteristic map for the holonomy groupoid of a foliation

Abstract

We prove a generalisation of Bott's vanishing theorem for the full transverse frame holonomy groupoid of any transversely orientable foliated manifold. As a consequence we obtain a characteristic map encoding both primary and secondary characteristic classes. Previous descriptions of this characteristic map are formulated for the Morita equivalent \'{e}tale groupoid obtained via a choice of complete transversal. By working with the full holonomy groupoid we obtain novel geometric representatives of characteristic classes. In particular we give a geometric, non-\'{e}tale analogue of the codimension 1 Godbillon-Vey cyclic cocycle of Connes and Moscovici in terms of path integrals of the curvature form of a Bott connection.