Square with edges, cube with faces identified.

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Spinnor
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Say I randomly give each edge of a square a direction and then I identify opposite edges, do I always come up with a two-dimensional compact manifold without boundary? Seems there are eight different edge orientation assignments, many being equivalent? How many different spaces?

Can I do the same with a cube? Give each edge of a cube a random orientation and identify opposing faces? Will we always be able to identify opposite faces with random edge orientation assignments? Can we say anything about such spaces, are any three-dimensional compact manifolds without boundary ?

Thanks for any help!
 

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Say I randomly give each edge of a square a direction and then I identify opposite edges, do I always come up with a two-dimensional compact manifold without boundary? Seems there are eight different edge orientation assignments, many being equivalent? How many different spaces?
Assuming your method of gluing does not allow corners, and that you always glue all pairs of edges, yes. The result can be simplified by referring to the classification theorem for surfaces without boundary. Since you are only using one square, the result is either a sphere, a torus, a projective plane, or the connected sum of 2 projective planes (the Klein bottle), up to homeomorphism.

Can I do the same with a cube? Give each edge of a cube a random orientation and identify opposing faces? Will we always be able to identify opposite faces with random edge orientation assignments? Can we say anything about such spaces, are any three-dimensional compact manifolds without boundary ?

Thanks for any help!
Classification of these objects is still open. See http://www.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf .
 

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