Hantzsche–Wendt manifold

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 Hantzsche–Wendt manifold is made from two cubes stacked on top of each other by gluing the identically-marked faces such that the markings line up.[5]

The HW manifold can be built from two cubes that share a face. One construction proceeds as follows:[5]

  1. The top and bottom faces are glued to one another.
  2. One of the remaining sides is glued to the opposite side with a 180° rotation.
  3. 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.
  4. 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[1]
Dimension Total
(OEIS: A124153)		 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.

References
  1. Rossetti, Juan P.; Szczepański, Andrzej (2005). "Generalized Hantzsche-Wendt flat manifolds". Revista Matemática Iberoamericana. 21 (3): 1053–1070. arXiv:math/0208205. doi:10.4171/RMI/445. S2CID 8222899.
  2. Hantzsche, Walter; Wendt, Hilmar (1935). "Dreidimensionale euklidische Raumformen". Mathematische Annalen. 110 (1): 593–611. doi:10.1007/bf01448045. ISSN 0025-5831. S2CID 119961050.
  3. Miatello, R. J.; Rossetti, J. P. (29 October 1999). "Isospectral Hantzsche-Wendt manifolds". Journal für die reine und angewandte Mathematik (Crelle's Journal). 1999 (515): 1–23. doi:10.1515/crll.1999.077. ISSN 1435-5345.
  4. Aurich, R; Lustig, S (21 August 2014). "The Hantzsche–Wendt manifold in cosmic topology". Classical and Quantum Gravity. 31 (16): 165009. arXiv:1403.2190. Bibcode:2014CQGra..31p5009A. doi:10.1088/0264-9381/31/16/165009. ISSN 0264-9381. S2CID 119223504.
  5. Demianski, Marek (2003). "Topology of the universe and the cosmic microwave background radiation". In Sanchez, Norma; Parijskij, Yuri (eds.). The early universe and the cosmic microwave background: theory and observations. Dordrecht: Kluwer Academic Publishers. pp. 166–169. ISBN 978-1-4020-1800-8.
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