Foliation groupoids and their cyclic homology

Abstract

The purpose of this paper is to prove two theorems which concern the position of etale groupoids among general smooth or Lie groupoids Our motivation comes from the noncommutative geometry and algebraic topology concerning leaf spaces of foliations Here one is concerned with invariants of the holonomy groupoid of a foliation such as the cohomology of its classifying space the cyclic homology of its smooth convolution algebra or the Ktheory of the Cconvolution algebras Many results here depend on the fact that such a holonomy groupoid can be reduced to what is called a complete transversal of the foliation giving rise to an equivalent etale groupoid For etale groupoids sometimes called rdiscrete groupoids in the literature

the cyclic homology sheaf theory and classifying spaces are each well understood as is the relation between these