The theorem of classification of closed surfaces says that any closed surface is homeomorphic to a fundamental polygon in the plane.(adsbygoogle = window.adsbygoogle || []).push({});

I was wondering if any fundamental polygon can be made into a closed surface by adjoining an appropriate atlas to it.

The topological requirements of a closed surface (compactness and connectedness) are certainly met by the fund. polygon but can we give it an atlas, and if so, is the resulting 2-manifold boundaryless?

Sure as hell appears so to me but I want to make sure.

Side question: Say we have a polygon with sides identified, and say that some side "a" is identified with n sides. Is n a topological invariant?

Thx!

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# Fundamental polygons and surfaces

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