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?(adsbygoogle = window.adsbygoogle || []).push({});

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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# Square with edges, cube with faces identified.

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