Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, by Marius Crainic

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.

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 [WeXu]).

As a second application we extend van Est's argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately gives a new proof, and a slight improvement of Hector-Dazord's integrability criterion [DaHe].

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 theta_L [ELW] (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles [BiLo, KT].

In the last section we describe some applications to Poisson geometry.

Marius Crainic <crainic@math.uu.nl>