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arXiv · 2507.05425

A Low-Dimensional Counterexample to the HK-Conjecture

Rachel Chaiser
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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 $d\geq 4$ 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 $d\leq 3$ the HK-conjecture holds for the associated odometer. We also discuss implications for the stable and unstable groupoid of a Smale space.

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Rachel Chaiser. 2025-07-07. A Low-Dimensional Counterexample to the HK-Conjecture. https://arxiv.org/abs/2507.05425

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