# Topological spaces made by identifying opposite faces of a cube?

My bashful, nameless, colleague asked me:

When you identify opposite faces of a square, then depending on where you twist or not, you get a torus, Klein bottle, or projective plane.

What spaces can you get when identifying opposite faces of a cube?

He was hoping for a reference.

• Of course, if you do the same with a dodecahedron with the correct twist, you famously get the Poincare homology sphere! en.wikipedia.org/wiki/… Feb 4 '14 at 1:01