A Low-Dimensional Counterexample to the HK-Conjecture

2025/07/07 by Rachel Chaiser, Chaiser, Rachel
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2507.05425

Abstract

We provide a counterexample to the HK-conjecture using the flat manifold odometers constructed by Deeley. Deeley's counterexample uses an odometer built from a flat manifold of dimension 9 and an expansive self-cover. We strengthen this result by showing that for each dimension d≥ 4 there is a counterexample to the HK-conjecture built from a flat manifold of dimension d. Moreover, we show that this dimension is minimal, as if d≤ 3 the HK-conjecture holds for the associated odometer. We also discuss implications for the stable and unstable groupoid of a Smale space.

Citations

• Homology and K-theory of dynamical systems. IV. Further structural results on groupoid homology
• Invariants for the Smale space associated to an expanding endomorphism of a flat manifold
• Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces
• A counterexample to the HK-conjecture that is principal
• Ample groupoids: equivalence, homology, and Matui's HK conjecture
• Homology of the Katsura-Exel-Pardo groupoid
• Nuclear dimension and classification of C*-algebras associated to Smale spaces
• Etale groupoids arising from products of shifts of finite type
• Dynamic Asymptotic Dimension: relation to dynamics, topology, coarse geometry, and C^*-algebras
• Classification of real Bott manifolds and acyclic digraphs

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