Haefliger's differentiable cohomology

2020/12/14 by Accornero, Luca, Crainic, Marius
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.07777

Abstract

We review Haefliger's differentiable cohomology for the pseudogroup of diffeomorphisms of 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.

Related

• Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes
• Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry
• Morphisms between complete Riemannian pseudogroups
• Chern-Weil theory for Haefliger-singular foliations
• Hopf cyclic cohomology and transverse characteristic classes