# 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:

The **Klein Bottle** has the associated group word:

And the **Projective Plane** has the associated group word: