Poincaré homology 3-sphere

Each dodecahedron in the 120-cell is a fundamental domain for the action of Γ on SU(2).

The quotient space SU(2)/Γ is called the Poincaré homology sphere.
It has the homology of S³, but its fundamental group is Γ.

We can also build it by taking one dodecahedron and identifying its opposite faces with the smallest possible clockwise twist.

For more, read the Wikipedia article homology sphere and especially this article:

Robion Kirby and Martin Scharlemann, Eight faces of the Poincaré homology 3-sphere, Uspekhi Matematicheskikh Nauk 37 (1982), 139–159.