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

TL;DR

This paper advances the understanding of the homology of étale groupoids from dynamical systems by establishing new formulas, duality results, and computing examples, including proving the HK conjecture for certain non-ample groupoids.

Contribution

It introduces a Künneth formula and Poincaré duality for specific groupoids, and verifies the HK conjecture for systems associated with algebraic number solenoids.

Findings

Proved a Künneth formula for product groupoids.

Established a Poincaré-duality type result for principal groupoids.

Confirmed the HK conjecture for certain non-ample groupoids.

Abstract

We consider the homology theory of \'etale groupoids introduced by Crainic and Moerdijk, with particular interest to groupoids arising from topological dynamical systems. We prove a K\"unneth formula for products of groupoids and a Poincar\'e-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.