A Baum–Connes conjecture for singular foliations

Abstract

We consider singular foliations whose holonomy groupoid may be nicely decomposed using Lie groupoids (of unequal dimension). We construct a $K$-theory group and a natural assembly type morphism to the $K$-theory of the foliation $C^{\ast }$-algebra generalizing to the singular case the Baum–Connes assembly map. This map is shown to be an isomorphism under assumptions of amenability. We examine some simple examples that can be described in this way and make explicit computations of their $K$-theory.