The Hantzsche–Wendt manifold, also known as the HW manifold or didicosm,
is a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
,
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
,
flat
Flat or flats may refer to:
Architecture
* Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries
Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
, first studied by
Walter Hantzsche and Hilmar Wendt in 1934. It is the only closed flat 3-manifold with first
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplici ...
zero. Its
holonomy group
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
is
.
It has been suggested as a possible
shape of the universe because its complicated geometry can obscure the features in the
cosmic microwave background
In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
that would arise if the universe is a closed
flat manifold In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles o ...
, such as the
3-torus
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, \mathbb^3 = S^1 \times S^1 \times S^1. In contrast, the usual torus is the Cartesian product of only two ...
.
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.
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.
The number of orientable HW manifolds up to
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ...
increases exponentially with dimension.
All of these have first Betti number ''β''
1 0 or 1.
References
{{DEFAULTSORT:Hantzsche-Wendt manifold
3-manifolds