# Hantzsche-Wendt manifolds

Recently a preprint was put on the arXiv:2103.01051 about Hantzsche-Wendt manifolds. I did not know what these manifolds are, got interested and started reading the introduction.

By definition, which goes back to arXiv:math/0208205, a Hantzsche-Wendt manifold is an orientable, n-dimensional flat manifold whose holonomy group is an elementary abelian 2-group of rank n-1, i.e., isomorphic to $$(\mathbb{Z}/2\mathbb{Z})^{n-1}$$.

Every n-dimensional flat manifold $$X$$ is a quotient $$\mathbb{R}^n / \Gamma$$, where $$\Gamma$$ is a Bieberbach group, i.e., a torsion-free, co-compact and discrete subgroup of $$\mathrm{Isom}(\mathbb{R}^n) \cong \mathbb{R}^n \rtimes \mathrm{O}(n)$$. The group $$\Gamma$$ fits into a short exact sequence $0 \to \mathbb{Z}^n \to \Gamma \to G \to 0\,,$ where the image of $$\mathbb{Z}^n$$ in $$\Gamma$$ is its maximal abelian normal subgroup and $$G$$ is finite and coincides with the holonomy group of $$X$$.

Hantzsche-Wendt manifolds exist in every odd dimension at least three, and since they are flat they are aspherical (their universal cover is contractible since it is $$\mathbb{R}^n$$). They also have some more interesting properties:

edit (April 14th, 2021): The (unique) three-dimensional Hantzsche-Wendt manifold has another interesting property. Its associated Bieberbach group $$\Gamma$$ is the first counterexample to the unit conjecture (https://blog.spp2026.de/unit-conjecture-disproved/)!