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

2023/10/15 by Valerio Proietti, Makoto Yamashita, Proietti, Valerio +1 · 1 citation
Mathematics · #Advanced Topics in Algebra #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2310.09928

openalex publication_date 2023/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• A Low-Dimensional Counterexample to the HK-Conjecture

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