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

Abstract

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.

v4: 23 pages, post-publication notes; v3: 23 pages, new title, Smale space part is split off as arXiv:2104.10938, to appear in Ergodic Theory Dynam. Systems; v2: 38 pages, some remarks and appendix expanded; v1: 34 pages

References in corpus (1)

• Renault's Equivalence Theorem for Groupoid Crossed Products

Cited by in corpus (6)

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• Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces
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• A geometric Elliott invariant and noncommutative rigidity of mapping tori