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

Abstract

In the rst section we discuss Morita invariance of dierentiablealgebroid cohomology In the second section we present an extension of the van Est isomorphism to groupoids As a rst application we clarify the connection between dierentiable and algebroid cohomology proved in degree and conjectured in degree by WeinsteinXu

As a second application we extend van Ests argument for the integrability of Lie algebras Applied to Poisson manifolds this immediately gives a slight improvement of Hector Dazords 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 EvensLuWeinsteins characteristic class L

hence in particular the modular class of Poisson manifolds and also the classical characteristic classes of at vector bundles

In the last section we describe some applications to Poisson geometry