Identifying Polygons and Their Quotient Spaces
Consider the torus $T^2$. Recall that this object can be obtained as the quotient space of a filled square with opposite edges identified in the same orientation as a quotient space:
Given a filled polygon with $2n$ edges, we can actually construct many different spaces by identifying pairs of edges in certain orientations. We define these polygons below.
|Definition: The Identifying Polygon is the filled polygon with $2n$ edges and their orientations identified to obtain a quotient space.|
The following remarkable theorem tells us that every connected compact 2-manifold is homeomorphic to one of these identifying polygons.
|Theorem 1: Every connected compact 2-manifold is homeomorphic to a quotient space arising from an identifying polygon with $2n$ edges whose edges are identified in pairs.|
We can associate every identifying polygon with a group word which describes the order and the orientation of the edges of the polygon. For example, the Torus above has the associated group word:(1)
The Klein Bottle has the associated group word:(2)
And the Projective Plane has the associated group word:(3)