Hantzsche–Wendt manifold explained
The Hantzsche–Wendt manifold, also known as the HW manifold or didicosm,[1] is a compact, orientable, flat 3-manifold, first studied by Walter Hantzsche and Hilmar Wendt in 1934.[2] It is the only closed flat 3-manifold with first Betti number zero. Its holonomy group is
.
[3] It has been suggested as a possible
shape of the universe because its complicated geometry can obscure the features in the
cosmic microwave background that would arise if the universe is a closed
flat manifold, such as the
3-torus.
[4] Construction
The HW manifold can be built from two cubes that share a face. One construction proceeds as follows:
- The top and bottom faces are glued to one another.
- One of the remaining sides is glued to the opposite side with a 180° rotation.
- One of the remaining faces on the top cube is glued to the matching face of the bottom cube, reflected across an axis parallel to the long axis of the double-cube.
- Repeat step 3 for the remaining pair of faces.
Generalizations
In addition to the orientable one (the Hantzsche–Wendt manifold), there are two non-orientable flat 3-manifolds with holonomy group
, known as the
first and
second amphidicosms, both with first Betti number 1.
[1] Similar flat n-dimensional manifolds with holonomy
, known as
generalized Hantzsche–Wendt manifolds, may be constructed for any
n≥2, but orientable ones exist only in odd dimensions.
[1] The number of orientable HW manifolds up to
diffeomorphism increases exponentially with dimension.
[3] All of these have first Betti number
β1 0 or 1.
[1] Number of generalized HW manifolds by dimension| Dimension | Total
| Orientable | Non-orientable | β1 = 0 | β1 = 1 |
|---|
| 2 | 1 | 0 | 1 | 0 | 1 |
| 3 | 3 | 1 | 2 | 1 | 2 |
| 4 | 12 | 0 | 12 | 2 | 10 |
| 5 | 123 | 2 | 121 | 23 | 100 |
| 6 | 2536 | 0 | 2536 | 352 | 2184 | |
Trivia
The didicosm is eponymous and plays a central role in Greg Egan's science-fiction short story "Didicosm".
Notes and References
- Rossetti . Juan P. . Szczepański . Andrzej . Generalized Hantzsche-Wendt flat manifolds . Revista Matemática Iberoamericana . 2005 . 21 . 3 . 1053–1070 . 10.4171/RMI/445 . math/0208205. 8222899 .
- Hantzsche. Walter. Wendt. Hilmar. 1935. Dreidimensionale euklidische Raumformen. Mathematische Annalen. 110. 1. 593–611. 10.1007/bf01448045. 119961050 . 0025-5831.
- Miatello . R. J. . Rossetti . J. P. . Isospectral Hantzsche-Wendt manifolds . Journal für die reine und angewandte Mathematik (Crelle's Journal) . 29 October 1999 . 1999 . 515 . 1–23 . 10.1515/crll.1999.077 . en . 1435-5345.
- Aurich . R . Lustig . S . The Hantzsche–Wendt manifold in cosmic topology . Classical and Quantum Gravity . 21 August 2014 . 31 . 16 . 165009 . 10.1088/0264-9381/31/16/165009 . 1403.2190 . 2014CQGra..31p5009A . 119223504 . 0264-9381.
