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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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  • $\begingroup$ 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/… $\endgroup$ – Russ Woodroofe Feb 4 '14 at 1:01
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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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  • $\begingroup$ Many good answers, but I guess this is the most directly related one. $\endgroup$ – Allen Knutson Mar 6 '12 at 18:11
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  • B. Everitt. 3-manifolds from platonic solids. Topology and its applications, 2004.

Covers everything you're asking for and more.

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  • $\begingroup$ This is wonderful! $\endgroup$ – Dylan Wilson Mar 6 '12 at 6:14
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Chapter 4 of "Topology Now!" by Messer & Straffin gives a good undergraduate level overview of the topic of gluing polyhedral solids.

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You (or the bashful colleague) might want to look at Cannon/Floyd/Parry's program to analyze spaces of this ilk: http://www.math.vt.edu/people/floyd/research/software/twist.html

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