Seifert-Weber space

In mathematics, Seifert-Weber space is a closed hyperbolic 3-manifold. It is also known as Seifert-Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds.

To construct it, notice that each face of a dodecahedron has an opposite face. We will glue each face to its opposite in a manner to get a closed 3-manifold. There are three ways to do this consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert-Weber space. The edges of the original dodecahedron meet in fives, making the dihedral angle 72° rather than 117° as in the regular dodecahedron in Euclidean space. The Seifert-Weber space is thus congruent to one cell of the Order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space. Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space.

The Seifert-Weber space is a rational homology sphere, and its first homology group is isomophic to $mathbb\; Z\_5^3$. It's conjectured that the Seifert-Weber space is not a Haken manifold -- ie: it does not contain any incompressible surfaces.