2003/12/31 by Marius Crainic · 9 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry #Advanced Topics in Algebra
paper · pdf · doi:10.1007/s00014-001-0766-9
openalex publication_date 2003/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/17
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 θL [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.