A Low-Dimensional Counterexample to the HK-Conjecture
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913932034179072 |
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| author | Chaiser, Rachel |
| author_facet | Chaiser, Rachel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_05425 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Low-Dimensional Counterexample to the HK-Conjecture Chaiser, Rachel Operator Algebras Geometric Topology K-Theory and Homology 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. |
| title | A Low-Dimensional Counterexample to the HK-Conjecture |
| topic | Operator Algebras Geometric Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2507.05425 |