Differentiable and algebroid cohomology , van Est isomorphisms , and characteristic classes ∗

In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we extend the Van Est isomorphism to groupoids. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and conjectured in degree 2 by Weinstein-Xu [50]). As a second application we extend Van Est’s argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately implies the integrability criterion of Hector-Dazord [14]. In the third section we describe the relevant characteristic classes of representations, living in algebroid cohomology, as well as their relation to the Van Est map. This extends Evens-Lu-Weinstein’s characteristic class θL [20] (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles [2, 30]. In the last section we describe applications to Poisson geometry.