Question

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.

Answer 1

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

Answer 2

• B. Everitt. 3-manifolds from platonic solids. Topology and its applications, 2004.

Covers everything you're asking for and more.

Answer 3

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

Answer 4

Chapter 4 of "Topology Now!" by Messer & Straffin gives a good undergraduate level overview of the topic of gluing polyhedral solids.