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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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Do all combinatorially distinct fundamental polygons correspond to surfaces?

Great question! There's an interesting comment at https://math.stackexchange.com/questions/204405/for-an-n-gon-how-many-fundamental-polygons-are-there "we ... disallow discontinuities, [i.e] two A e …
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