# 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.

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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/… –  Russ Woodroofe Feb 4 at 1:01

The ones that are manifolds were considered by Poincaré, and a nice discussion is on this page of the Manifold Atlas.

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Many good answers, but I guess this is the most directly related one. –  Allen Knutson Mar 6 '12 at 18:11
• B. Everitt. 3-manifolds from platonic solids. Topology and its applications, 2004.

Covers everything you're asking for and more.

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This is wonderful! –  Dylan Wilson Mar 6 '12 at 6:14