Ample groupoids: Equivalence, homology, and Matui's HK conjecture
We investigate the homology of ample Hausdorff groupoids. We establish that a number of notions of equivalence of groupoids appearing in the literature coincide for ample Hausdorff groupoids, and deduce that they all preserve groupoid homology. We compute the homology of a Deaconu–Renault groupoid a...
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Dokumenttypen: | Artikel |
Medientypen: | Text |
Erscheinungsdatum: | 2019 |
Publikation in MIAMI: | 24.10.2019 |
Datum der letzten Änderung: | 28.10.2019 |
Quelle: | Münster Journal of Mathematics, 12 (2019), S. 411-451 |
Verlag/Hrsg.: |
Mathematisches Institut (Universität Münster)
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Angaben zur Ausgabe: | [Electronic ed.] |
Fachgebiet (DDC): | 510: Mathematik |
Lizenz: | InC 1.0 |
Sprache: | Englisch |
Format: | PDF-Dokument |
URN: | urn:nbn:de:hbz:6-53149724313 |
Weitere Identifikatoren: | DOI: 10.17879/53149724091 |
Permalink: | https://nbn-resolving.de/urn:nbn:de:hbz:6-53149724313 |
Onlinezugriff: | mjm_2019_12_411-451.pdf |
We investigate the homology of ample Hausdorff groupoids. We establish that a number of notions of equivalence of groupoids appearing in the literature coincide for ample Hausdorff groupoids, and deduce that they all preserve groupoid homology. We compute the homology of a Deaconu–Renault groupoid associated to k pairwise-commuting local homeomorphisms of a zero-dimensional space, and show that Matui’s HK conjecture holds for such a groupoid when k is one or two. We specialize to k-graph groupoids, and show that their homology can be computed in terms of the adjacency matrices, using a chain complex developed by Evans. We show that Matui’s HK conjecture holds for the groupoids of single vertex k-graphs which satisfy a mild joint-coprimality condition. We also prove that there is a natural homomorphism from the categorical homology of a k-graph to the homology of its groupoid.