Haefliger’s differentiable cohomology

We review Haefliger’s differentiable cohomology for the pseudogroup of diffeomorphisms of \mathbb{R}^{q} ; see “Haefliger (1976)”. We unravel the structure that governs such cohomologies, which, remarkably, is related to the so called Cartan distribution underlying the geometric study of PDEs. Hence, we extend Haefliger’s differentiable cohomology to the general framework of 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 associated with flat Cartan groupoids. The outcome generalizes the existing approaches to characteristic classes for foliations “Bernšteĭn and Rosenfel’d (1972)”, “Bott and Haefliger (1972)”, “Bernšteĭn and Rosenfel’d (1973)”, and “Haefliger (1976)”. The motivation for this work is two-fold. On the one hand, it is motivated by the recent approach to geometric structures via multiplicative (Cartan) distributions; see “Salazar (2013)”, “Yudilevich (2016)”, and “Cattafi (2020)”; from that perspective, we are constructing characteristic classes for such structures. On the other hand, it is motivated by our (ongoing) attempt to turn classical symmetries (pseudogroups) into non-commutative, Hopf-algebraic, ones; such attempt is inspired by existing work in non-commutative geometry; see “Connes and Moscovici (2001)”, “Moscovici and Rangipour (2009)”, and “Moscovici and Rangipour (2011)”. It also aims at a unified approach which allows for non-transitive pseudogroups.