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A surface is a two-dimensional topological manifold. The term can also be used to describe a smooth surface, depending on the context.

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"Classification" of (orientable) 3-manifolds with genus-g-surface as their boundary

This is in some sense a generalization of the question I asked some time ago. I am very sorry if this question is too basic for MathOverflow, but I just started learning about some more detailed thing …