Proof of a conjecture of A. Haefliger
type: Item
instance of: scholarly article
title: Proof of a conjecture of A. Haefliger
zbMATH Open document ID: 0897.22003
DOI: 10.1016/S0040-9383(97)00053-0
published in: Topology
publication date: 1998-10-05T00:00:00Z
review text: An étale groupoid is a topological groupoid whose source and target maps are local homeomorphisms. The main examples of these groupoids are the Haefliger groupoid \(\Gamma^{q}\) and the holonomy groupoid of a foliation. Each étale groupoid \(G\) defines an abelian category on \(G\)-sheaves. If \(A\) is an arbitrary \(G\)-sheaf, we can consider a natural sheaf cohomology \(H^{*} (G,A)\) with coefficients in \(A\) and also a sheaf \(\widetilde{A}\) on a classifying space \(BG\) of \(G\). In this paper, the author clarifies the above concepts and proves the following result conjectured by Haefliger: There exists an isomorphism \[ H^{*} (G,A) \cong H^{*} (BG,\widetilde{A}) \] natural in \(G\) and \(A\). Finally, the factorial properties of \(H^{*} (G,A)\) are outlined in an appendix.
Mathematics Subject Classification ID: 22A30
Mathematics Subject Classification ID: 22A22
Mathematics Subject Classification ID: 20L05
Mathematics Subject Classification ID: 55N30
zbMATH DE Number: 1133062
zbMATH Keywords: topological groupoids
zbMATH Keywords: groupoid of a foliation
zbMATH Keywords: sheaves on groupoids
zbMATH Keywords: étale groupoid
zbMATH Keywords: Haefliger groupoid
Wikidata QID: Q123082105
author: Ieke Moerdijk
reviewed by: Carlos Galindo
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1133062