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.



Topological spaces made by identifying opposite faces of a cube?
The ones that are manifolds were considered by Poincaré, and a nice discussion is on [this page of the Manifold Atlas](http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_cube_manifolds).


