Examples of Classifying Connected Compact 2-Manifolds 1

Example 1

Determine which connected compact $2$-manifold is given by the following identifying polygon:

We perform some cut and pastes as illustrated below:

The final identifying polygon is an identifying polygon for the projective plane.

An alternative way to see that the identifying polygon above corresponds to the projective plane is to let $d = abc$. Then this identifying polygon corresponds to the group word $d^2 = 1$, so it corresponds to the projective plane.