Haefliger's differentiable cohomology

TL;DR

This paper explores Haefliger's differentiable cohomology, extending it to flat Cartan groupoids, and relates it to geometric structures and characteristic classes, providing a broader framework for foliation theory.

Contribution

It introduces a new cohomology theory for flat Cartan groupoids and connects it to existing foliation characteristic classes, generalizing previous approaches.

Findings

Defined an analogue of Haefliger cohomology for flat Cartan groupoids

Established a Van Est-like map relating infinitesimal and global cohomology

Developed a characteristic map for geometric structures on manifolds

Abstract

We review Haefliger's differentiable cohomology for the pseudogroup of diffeomorphisms of $\mathbb{R}^q$. We investigate the structure needed to define such a cohomology, which, remarkably, is related to the so called Cartan distribution underlying the geometric study of PDE. We define an analogue of Haefliger differentiable cohomology for flat Cartan groupoids, investigate its infinitesimal counterpart and relate the two by a Van Est-like map. Finally, we define a characteristic map for geometric structures on manifolds $M$ associated to flat Cartan groupoids. The outcome generalizes the existing approaches to characteristic classes for foliations.