Hantzsche–Wendt manifold

[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] The HW manifold can be built from two cubes that share a face.

One construction proceeds as follows:[5] In addition to the orientable one (the Hantzsche–Wendt manifold), there are two non-orientable flat 3-manifolds with holonomy group

, 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.

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 ]