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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

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 "anticlockwise". A space is o ...

, flat

3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

, first studied by

Walter Hantzsche Walter may refer to: People and fictional characters * Walter (name), including a list of people and fictional and mythical characters with the given name or surname * Little Walter, American blues harmonica player Marion Walter Jacobs (1930–19 ...

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 simplicia ...

zero. Its

holonomy group In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence ...

\mathbb_2^2

. It has been suggested as a possible

shape of the universe In physical cosmology, the shape of the universe refers to both its local and global geometry. Local geometry is defined primarily by its curvature, while the global geometry is characterised by its topology (which itself is constrained by curv ...

because its complicated geometry can obscure the features in the

cosmic microwave background The cosmic microwave background (CMB, CMBR), or relic radiation, is microwave radiation that fills all space in the observable universe. With a standard optical telescope, the background space between stars and galaxies is almost completely dar ...

that would arise if the universe is a closed flat manifold, 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

\mathbb_2^2

, known as the first and second amphidicosms, both with first Betti number 1. Similar flat ''n''-dimensional manifolds with holonomy

\mathbb_2^

, 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 differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

increases exponentially with dimension. All of these have first Betti number ''β''₁ 0 or 1.

Trivia

The didicosm is eponymous and plays a central role in

Greg Egan Greg Egan (born 20 August 1961) is an Australian science fiction writer and mathematician, best known for his works of hard science fiction. Egan has won multiple awards including the John W. Campbell Memorial Award, the Hugo Award, and the Lo ...

science-fiction Science fiction (often shortened to sci-fi or abbreviated SF) is a genre of speculative fiction that deals with imaginative and futuristic concepts. These concepts may include information technology and robotics, biological manipulations, sp ...

short story A short story is a piece of prose fiction. It can typically be read in a single sitting and focuses on a self-contained incident or series of linked incidents, with the intent of evoking a single effect or mood. The short story is one of the old ...

" Didicosm".

References

{{DEFAULTSORT:Hantzsche-Wendt manifold 3-manifolds