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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.

11 votes

Morse theory in TOP and PL categories?

Forman's papers and the books by Kozlov and Orlik-Welker that you see cited at Daniel's Wikipedia link are good starting points for the more combinatorial tradition of PL Morse theory. For more geom …
Sergey Melikhov's user avatar