Homology and -theory of dynamical systems III. Beyond stably disconnected Smale spaces

We study homological invariants of étale groupoids arising from Smale spaces, continuing our previous work, but going beyond the stably disconnected case by incorporating resolutions in the space direction. We show that the homology groups defined by Putnam are isomorphic to the Crainic–Moerdijk groupoid homology with integer coefficients. We also show that the K K -groups of C ∗ ^* -algebras of stable and unstable equivalence relations have finite rank. For unstably disconnected Smale spaces, we provide a cohomological spectral sequence whose second page shows Putnam’s (stable) homology groups, and converges to the K K -groups of the unstable C ∗ ^* -algebra.