A going-down principle for {\'e}tale groupoids and the Baum-Connes conjecture

TL;DR

This paper extends the going-down principle to étale groupoids, providing new tools for analyzing the Baum-Connes conjecture, and demonstrates its applications in proving split injectivity and continuity results in topological K-theory.

Contribution

It introduces a generalized going-down principle for étale groupoids, expanding previous results and connecting bicategorical functoriality with the Baum-Connes conjecture.

Findings

Proves split injectivity of Baum-Connes assembly map for strongly amenable étale groupoids.

Establishes continuity of topological K-theory for étale groupoids.

Analyzes scope of Künneth formulas in the context of étale groupoids.

Abstract

We study a going-down principle for {\'e}tale groupoids and its applications, extending the earlier results for locally compact groups by Chabert, Echterhoff and Oyono-Oyono, and for ample groupoids by B{\"o}nicke and by B{\"o}nicke-Dell'Aiera. The proof in the general {\'e}tale groupoid setting is based on a more detailed study of groupoid simplicial complexes. We also study a bicategorical functoriality involving the induction functors from {\'e}tale groupoid correspondences, which was introduced by Miller. This yields a bicategorical interpretation of the induction-restriction adjunction. As an application of the going-down principle, we provide a proof of the split injectivity of Baum-Connes assembly map for {\'e}tale groupoids that are strongly amenable at infinity, recovering a result obtained by B{\"o}nicke and Proietti via a categorical approach. The going-down principle is also applied on the proof of continuity of topological K-theory of {\'e}tale groupoids and the study of scope of validity of K{\"u}nneth formulas.