I'm reading the book 'low-dimensional geometry from euclidean surfaces to hyperbolic knots' written by Francis Bonahon, and the it says that gluing euclidean polygon is homeomorphic to some surface without any proof. For example, the quotient space made by gluing opposite sides of euclidean rectangle is homeomorphic to torus, or the quotient space made by gluing opposite sides of hexagon is homeomorphic to double torus. I tried to understand this homeomorphism by trivial continuous bijection map (which is closed map since the polygon is compact). But it doesn't work to deal with hyperbolic polygon since if there is any vertical line geodesic (infinite radius with center on infinite) making the polygon, it implies that the polygon is not compact. How can I understand the homeomorphism btw the quotient space made by gluing the sides of euclidean(hyperbolic, spherical) polygon and the surface made by stretching the polygon. Thank you!
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1Does this answer your question? Prove that quotient map of polygon is surface – Moishe Kohan Jan 02 '21 at 14:48
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No, what I’m asking is that how to prove the quotient space is homeomorphic to the surface made by stretching the metric, not to show that it is actually a surface. Sorry. – 51 JUNJO Jan 03 '21 at 17:57
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I have no idea what "stretching a metric" means. You should edit your question to clarify it. As written, the first sentence is about lack of a proof in Bonahon's book and there is nothing about "stretching" there. – Moishe Kohan Jan 03 '21 at 18:00