nLab Hantzsche-Wendt manifold

Definition

The Hantzsche-Wendt manifold (also called HW manifold or didicosm) is the coset space

$$D \coloneqq T^3/(\mathbb{Z}_2\times\mathbb{Z}_2) \,,$$

namely the quotient space of the 3-torus $T^3$ by the antipodal action of $\mathbb{Z}_2\times\mathbb{Z}_2$ on two components. It is a compact, orientable and flat 3-manifold.

Properties

• The holonomy group of the Hantzsche-Wendt manifold is $\mathbb{Z}_2^2$. (Miatello & Rossetti 99)
• The first homology group of the Hantzsche-Wendt manifold is $\mathbb{Z}_4^2$. (Conway & Rossetti 03, p. 23, Egan 23)

Generalizations

The first and second amphidicosm are compact, non-orientable and flat $3$-manifolds with holonomy group $\mathbb{Z}_2^2$ and first Betti number $1$.

A generalized Hantzsche-Wendt manifold is a compact and flat $n$-manifold with holonomy group $\mathbb{Z}_2^{n-1}$.

References

First studied in:

• Walter Hantzsche, Hilmar Wendt, Dreidimensionale euklidische Raumformen, Mathematische Annalen. 110 (1935) 1, p. 593–611, doi:10.1007/bf01448045. ISSN 0025-5831, S2CID 119961050

General:

• John Conway and Jean-Paul Rossetti, Describing the platycosms (2003), arXiv:math/0311476
• Roberto Miatello and Jean-Paul Rossetti, Isospectral Hantzsche-Wendt manifolds, Journal für die reine und angewandte Mathematik (Crelle’s Journal). 1999. Jahrgang, Nr. 515, 29. Oktober 1999, ISSN 1435-5345, S. 1–23, doi:10.1515/crll.1999.077

On coverings:

• Grigory Chelnokov and Alexander Mednykh, On the coverings of Hantzsche-Wendt manifold (2020), arXiv:2009.06691

Application in cosmic topology:

• Ralf Aurich and Sven Lustig, The Hantzsche-Wendt Manifold in Cosmic Topology (2014), arXiv:1403.2190

Explanation, visualisation and calculation of the first homology:

• Greg Egan, Loops Across Space (2023)

See also:

• Wikipedia, Hantzsche-Wendt manifold