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 Ests 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-Weinsteins characteristic class $\theta_{L}$ 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.