Proof of a conjecture of A. Haefliger

Abstract

This paper is concerned with the cohomology of etale groupoids. These are topological groupoids whose source and target maps are local homeomorphisms. Etale groupoids play a central r^ole in foliation theory (cf. e.g. [C, H84, BN, vE] and many others). Indeed, the main examples of etale groupoids include the Hae iger groupoid as well as the holonomy groupoid of any foliation. Any orbifold also gives rise to an etale groupoid in a natural way. Each etale groupoid G denes an abelian category of G-sheaves, and a natural sheaf cohomology H(G;A) with coecients in an arbitrary G-sheaf A. On the other hand, each such sheaf A also induces an (ordinary) sheaf ~ A on the classifying space BG of the groupoid G, and hence denes the usual sheaf cohomology groups H(BG; ~ A) of the space BG. I will prove the following theorem, conjectured by Hae iger (see e.g. [H76, H92]).