A Low-Dimensional Counterexample to the HK-Conjecture

TL;DR

This paper constructs low-dimensional counterexamples to the HK-conjecture using flat manifold odometers, demonstrating the conjecture's failure in dimensions four and higher, and confirming its validity in lower dimensions.

Contribution

It provides the first explicit counterexamples to the HK-conjecture in dimensions four and above, establishing minimal dimension thresholds and exploring implications for Smale spaces.

Findings

Counterexamples exist for all dimensions d ≥ 4.

The HK-conjecture holds for dimensions d ≤ 3.

Counterexamples are constructed from flat manifolds of dimension d.

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.