Homology and $K$-Theory of Torsion-Free Ample Groupoids and Smale Spaces

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 groupoid C*-algebra when the groupoid has torsion-free stabilizers and satisfies the strong Baum-Connes conjecture. The construction is based on the triangulated category approach to the Baum-Connes conjecture by Meyer and Nest. For the unstable equivalence relation of a Smale space with totally disconnected stable sets, this spectral sequence shows Putnam's homology groups on the second sheet.