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

TL;DR

This paper develops a spectral sequence linking groupoid homology to K-theory of groupoid C*-algebras for torsion-free ample groupoids, advancing understanding in operator algebras and dynamical systems.

Contribution

It introduces a spectral sequence connecting groupoid homology to K-theory for torsion-free ample groupoids, utilizing the Meyer-Nest triangulated category approach.

Findings

Spectral sequence converges to K-groups under specified conditions.

Applications to topological dynamics are demonstrated.

Discussion of the HK conject of Matui is included.

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.