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arXiv · 2602.19787

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

Kai Mao
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Abstract

We study a going-down principle for {\'e}tale groupoids and its applications, extending previous 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. For the most commonly considered kind of going-down functors, we recover the result of B{\"o}nicke and Proietti, which they proved via a categorical approach and used to establish the split injectivity of the Baum--Connes assembly map for {\'e}tale groupoids that are strongly amenable at infinity. We also study a bicategorical functoriality, involving the induction functors from {\'e}tale groupoid correspondences introduced by Miller. This yields a bicategorical interpretation of the induction-restriction adjunction. The going-down principle is also applied to the proof of continuity of topological K-theory of {\'e}tale groupoids and the study of the scope of validity of K{\"u}nneth formulas.

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BibTeXRIS

Kai Mao. 2026-02-23. A going-down principle for {\'e}tale groupoids and the Baum-Connes conjecture. https://arxiv.org/abs/2602.19787

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