Rachel Chaiser | Searcharxiv

arXiv2025

A Low-Dimensional Counterexample to the HK-Conjecture

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.

math.OA↗

arXiv2022

Invariants for the Smale space associated to an expanding endomorphism of a flat manifold

We study invariants associated to Smale spaces obtained from an expanding endomorphism on a (closed connected Riemannian) flat manifold. Specifically, the relevant invariants are the $K$-theory of the associated $C^*$-algebras and Putnam's homology theory for Smale spaces. The latter is isomorphic to the groupoid homology of the groupoids used to construct the $C^*$-algebras.

math.KT↗