I have a question which I just do not seem to see the answer for:
I am proving the classification theorem for compact surfaces and use planar diagrams as representation of the surfaces. I follow the combinatorical proof of Seifert and Threlfall with pasting and cutting on the planar diagrams. One of its steps says that the planar model gets reduced down to one single vertex.
Now, if I only have an algebraic representation of a planar model, e.g. bbcca^(-1)dda how do I know how many vertices there are? So far I think that if a vertex is the terminal point of an edge z, and the initial point of an ege y for example, then this must be ahered throught the whole planar diagram.
In this exact example above why cant it all be identified to one vertex? Or more specific, why cant any vertex out of bbcc be identified with one out of dd?
I am thankful for any help!