Valerio Proietti: Base change, groupoid homology, and hyperbolic dynamics

Abstract: In sheaf theory, base change theorems relate the direct image and the inverse image of sheaves. I will introduce a simple base change result in the context of ample groupoid modules and homology. As an application, I will explain that groupoid homology can recover Putnam's homology groups of Smale spaces. These are hyperbolic topological dynamical systems akin to the basic sets of Axiom A diffeomorphisms, orginally studied by Smale. The groupoids of Smale spaces produce several examples of interesting C*-algebras, and their K-theory groups are suitably approximated by groupoid homology. This approximation property can be used to show that K-theory groups of Smale space C*-algebras have finite rank (as conjectured by Kaminker, Putnam, and Whittaker), which in turn invalidates a conjecture that Smale spaces exhaust the range of K-theory on classifiable real rank zero C*-algebras.