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!