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:
- They are rational homology spheres https://doi.org/10.1112/S0025579300010561.
- They are cohomologically rigid over \(\mathbb{F}_2\), i.e., two Hantzsche-Wendt manifolds are homeomorphic if and only if their graded cohomology rings with coefficients in \(\mathbb{F}_2\) are isomorphic: https://doi.org/10.1016/j.aim.2016.08.004.
- There are pairs of Hantzsche-Wendt manifolds which are isospectral but not homeomorphic to each other: https://doi.org/10.1515/crll.1999.077.
- No Hantzsche-Wendt manifold of dimension at least five admits a spin\({}^c\)-structure: https://arxiv.org/abs/2103.01051. (But the only one in dimension three is actually even spin.)
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/)!