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

Marius Crainic
2003 Commentarii Mathematici Helvetici  
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 [47]). 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
more » ... improvement of Hector-Dazord's integrability criterion [12] . 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 [17] (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles [2, 26] . In the last section we describe some applications to Poisson geometry.
doi:10.1007/s00014-001-0766-9 fatcat:c75iqyuqivbxddfll7q5fdenba