Identifying Polygons and Their Quotient Spaces

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)
\begin{align} \quad aba^{-1}b^{-1} = 1 \end{align}

The Klein Bottle has the associated group word:

(2)
\begin{align} \quad aba^{-1}b = 1 \end{align}

And the Projective Plane has the associated group word:

(3)
\begin{align} \quad abab = 1 \end{align}