Abstract

The differentiable cohomology of a pseudogroup on a manifold M was defined by André Haefliger in the seventies, in the development of characteristic classes of foliations. In Haefliger's approach to the subject, foliations are seen as cocycles valued in the étale groupoid Gamma^q of germs of diffeomorphisms of R^q. From this point of view, the theory of characteristic classes for foliations is reminiscent of the theory of characteristic classes for flat principal g-bundles, which can be represented as cocycles valued in the discrete group G^delta. For such bundles, a "geometric" characteristic map is defined. It is a map from the relative cohomology H*(g, K) of the Lie algebra g of G to the cohomology of M. The classical Van-Est isomorphism for Lie groups allows one to interpret it as a map from the differentiable cohomology H_d*(G) of G to H*(M). A similar "geometric" map is available for foliations and is defined from the cohomology GF^q of the Gelfand-Fuchs Lie algebra of formal vector fields on R^q relative to O(q). Haefliger defined a differentiable complex for the groupoid Gamma^q and proved a Van Est isomorphism between the differentiable cohomology H*_{diff}(Gamma^q) and the cohomology of the relative Gelfand-Fuchs Lie algebra, making the analogy with flat principal bundles complete. His construction focuses on the groupoid Gamma^q and, probably also because of the seemingly ad-hoc approach, did not receive much attention. Furthermore, it is not clear which is the structure that makes the definition work. Our main driving question is to find such a structure. The outcome is the structure of "flat Cartan groupoid" on the infinite jet groupoid J^inftyGamma^q, which is well known for the role that plays in the geometric theory of PDE. After reviewing Haefliger's approach and introducing flat Cartan groupoids, we will define their "Haefliger cohomology", investigate its infinitesimal counterpart and relate the two by a Van Est map. Finally, we will construct a characteristic map for cocycles valued in a flat Cartan groupoid, which reduces to the "geometric" map for foliations when the groupoid is J^inftyGamma^q. This is joint work with Marius Crainic.