I've been stuck for a while now on this step of the classification theorem proof in Introduction to Topological Manifolds by John Lee:
STEP 4: M admits a presentation in which all vertices are identified to a single point.
Where M is a connected compact surface.
I don't understand how could this be true. Take a (bounded) cylinder for example. It has the following presentation: $aba^{-1}c$. I don't see how all vertices could be mapped to the same point, that could only happen if we turn it into a torus (i.e. identify b and c together). The key step in the above proof seems to be that he's assuming that for every edge $b$ there is another edge $b$ or $b^{-1}$ with which it is identified; but again: that's not true in the case of the cylinder, not all edges are identified. The same problem happens if we consider the presentation of a mobius band for example and other surfaces as well. I've rechecked the definitions several times to see if I was missing something basic but I'm not able to see where I went wrong so I would appreciate any help on this.