A Baum-Connes conjecture for singular foliations

TL;DR

This paper extends the Baum-Connes conjecture to singular foliations with decomposable holonomy groupoids, demonstrating its validity under amenability and providing explicit K-theory computations for various examples.

Contribution

It formulates and proves the Baum-Connes conjecture for a class of singular foliations with decomposable holonomy groupoids, including explicit K-theory calculations.

Findings

Baum-Connes conjecture holds under amenability for these foliations

Explicit K-theory computations for several examples

Holonomy groupoids can be decomposed using Lie groupoids of varying dimensions

Abstract

We consider singular foliations whose holonomy groupoid may be nicely decomposed using Lie groupoids (of unequal dimension). We show that the Baum-Connes conjecture can be formulated in this setting. This conjecture is shown to hold under assumptions of amenability. We examine several examples that can be described in this way and make explicit computations of their K-theory.