Foliation groupoids and their cyclic homology
نویسنده
چکیده
The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the non-commutative geometry and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of the holonomy groupoid of a foliation [4, 34], such as the cohomology of its classifying space [14], the cyclic homology of its smooth convolution algebra [2, 7], or the K-theory of the C-convolution 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 étale groupoid . For étale groupoids (sometimes called r-discrete groupoids in the literature [30, 33]), the cyclic homology, sheaf theory and classifying spaces are each well understood, as is the relation between these.
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