A going-down principle for {é}tale groupoids and the Baum-Connes conjecture
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918350803697664 |
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| author | Mao, Kai |
| author_facet | Mao, Kai |
| contents | We study a going-down principle for {é}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{ö}nicke and by B{ö}nicke-Dell'Aiera. The proof in the general {é}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 {é}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 {é}tale groupoids that are strongly amenable at infinity, recovering a result obtained by B{ö}nicke and Proietti via a categorical approach. The going-down principle is also applied on the proof of continuity of topological K-theory of {é}tale groupoids and the study of scope of validity of K{ü}nneth formulas. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_19787 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A going-down principle for {é}tale groupoids and the Baum-Connes conjecture Mao, Kai K-Theory and Homology Operator Algebras We study a going-down principle for {é}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{ö}nicke and by B{ö}nicke-Dell'Aiera. The proof in the general {é}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 {é}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 {é}tale groupoids that are strongly amenable at infinity, recovering a result obtained by B{ö}nicke and Proietti via a categorical approach. The going-down principle is also applied on the proof of continuity of topological K-theory of {é}tale groupoids and the study of scope of validity of K{ü}nneth formulas. |
| title | A going-down principle for {é}tale groupoids and the Baum-Connes conjecture |
| topic | K-Theory and Homology Operator Algebras |
| url | https://arxiv.org/abs/2602.19787 |