Homology and K-theory of dynamical systems I. Torsion-free ample groupoids

Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C $^*$ -algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum–Connes conj...

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Published inErgodic theory and dynamical systems Vol. 42; no. 8; pp. 2630 - 2660
Main Authors PROIETTI, VALERIO, YAMASHITA, MAKOTO
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.08.2022
Subjects
Algebra
Dynamical systems
Homology
Mathematics
Original Article
46L85
Baum–Connes conjecture
19K35
37D99
groupoid homology
K-theory
spectral sequence
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Summary:Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C $^*$ -algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum–Connes conjecture. The construction is based on the triangulated category approach to the Baum–Connes conjecture developed by Meyer and Nest. We also present a few applications to topological dynamics and discuss the HK conjecture of Matui.
Bibliography:ObjectType-Article-1
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ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2021.50