A characteristic map for the holonomy groupoid of a foliation

TL;DR

This paper extends Bott's vanishing theorem to the full transverse frame holonomy groupoid of transversely orientable foliated manifolds, providing new geometric representatives of characteristic classes and a non-étale analogue of the Godbillon-Vey cyclic cocycle.

Contribution

It generalizes Bott's theorem to the full holonomy groupoid, enabling new geometric descriptions of characteristic classes beyond the étale case.

Findings

Established a characteristic map for the full holonomy groupoid.

Derived a geometric, non-étale version of the Godbillon-Vey cyclic cocycle.

Connected characteristic classes with path integrals of curvature forms.

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.