I am practicing qualifying exam problems and I am having trouble with the following question. Any help is greatly appreciated.
Let $P$ be a polygon with an even number of sides. Suppose that the sides are identified in pairs in any way whatsoever. Prove that the quotient space is a manifold.
A manifold is, in particular, a locally Euclidean space, meaning that every point has a neighborhood homeomorphic to a disk in $\mathbb{R}^n$. You are working with a surface $(n = 2)$.
Do you see how every point in the interior of the polygon has a neighborhood that is unchanged by the identifications.
What about points on an edge, but not a vertex? They have half-disk neighborhoods in the polygon. Assuming that the identifications are nice (as Ryan Budney points out), two of these half disks are fused along a diameter, so you get a disk.
What happens to the vertices of the polygons?