Commentarii Mathematici Helvetici


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Volume 78, Issue 4, 2003, pp. 681–721
DOI: 10.1007/s00014-001-0766-9

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

Marius Crainic[1]

(1) Department of Mathematics, University of Utrecht, P.O. Box 80010, 3508 TA, UTRECHT, NETHERLANDS

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 [50]). As a second application we extend Van Ests argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately implies the integrability criterion of Hector-Dazord [14]. 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-Weinsteins characteristic class $\theta_{L}$ [20] (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles [2, 30]. In the last section we describe applications to Poisson geometry.

Keywords: Groupoids, Lie algebroids, Van Est isomorphism, cohomology,characteristic classes, Poisson geometry

Crainic M. Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes. Comment. Math. Helv. 78 (2003), 681-721. doi: 10.1007/s00014-001-0766-9