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 present an extension of the van Est isomorphism to groupoids. This immediately implies a version of Haefliger's conjecture for differentiable cohomology. 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 gives (a slight improvement of) Hector-Dazord's integrability criterion. 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 $θ_{L}$ (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 some applications to Poisson geometry (e.g. we clarify the Morita invariance of Poisson cohomology, and of the modular class).