Wikipedia gives the following fundamental polygon for the real projective plane $\mathbb{R}\mathrm{P}^2$
The problem here is that the corners aren't identified to a single point (like in the fundamental polygon of the torus). I don't think this picture is correct. The group presentation resulting from this polygon was considered in a previous question, and the same conclusion was reached.
What puzzles me more is that Hatcher's Algebraic Topology lists the same polygon as the fundamental polygon for the real projective plane (chapter 2, page 102), but the correct version (with only two edges identified together) is listed in an earlier page (chapter 1, page 51).
My question: Can this fundamental polygon made to actually represent the real projective plane in the sense that all corners are identified to a point and the resulting group presentation is $\mathbb{Z}_2$?
Thank you.
Best Answer
If you're ok with the fundamental polygon being a polygon on a surface which isn't $\mathbb{R}^2$, then you can get a polygon with two edges on the sphere $S^2$.
The polygon is given by taking the northern hemisphere, and the edges are given by each a half of the equator, with the identification being the antipodal map on the circle defining the equator.
If you know how covering spaces work, this region is a fundamental region of the $2$-fold covering of the sphere on to the real projective plane.