Homology and K-theory of dynamical systems. IV. Further structural results on groupoid homology

Abstract

We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk, with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for groupoids which are principal with orbits homeomorphic to a Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

arXiv commentv2: accepted version, minor changes; v1: splitting off part of arXiv:2207.03118 as a separate paper

References in corpus (9)

• Homology and topological full groups of etale groupoids on totally disconnected spaces
• Ample groupoids: equivalence, homology, and Matui's HK conjecture
• Homology and K-theory of dynamical systems. I. torsion-free ample groupoids
• Homology of odometers
• Groups with Spanier-Whitehead duality
• Homology and K-theory of dynamical systems. II. Smale spaces with totally disconnected transversal
• Dynamical correspondences for Smale spaces
• A note on homology for Smale spaces
• Invariants for the Smale space associated to an expanding endomorphism of a flat manifold