Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes

Abstract

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]). 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. 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, (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles. In the last section we describe applications to Poisson geometry.