A Guide to the Classification Theorem for Compact Surfaces by Jean Gallier & Dianna Xu

Author:Jean Gallier & Dianna Xu
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

We may still have several faces. We claim that if there are at least two faces, then for every face, A, there is some face, B, such that B ≠ A, , and there is some edge, a, both in the boundary of A and in the boundary of B. If this was not the case, there would be some face, A, such that for every face, B, such that B ≠ A and , every edge, a, in the boundary of B does not belong to the boundary of A. Then, every inner edge, a, occurring in the boundary of A must have both of its occurrences in the boundary of A, and of course, every boundary edge in the boundary of A occurs once in the boundary of A alone. But then, the cell complex consisting of the face A alone and the edges occurring in its boundary would form a proper subsystem of K, contradicting the fact that K is connected.

Thus, if there are at least two faces, from the above claim and using (P2) − 1, we can reduce the number of faces down to one. It is a simple matter to check that no new vertices are introduced and that loops are unaffected.

Next, if some boundary contains two occurrences of the same edge, a, i.e., it is of the form, aXaY , where X, Y denote strings of edges, with X, Y ≠ ε, we show how to make the two occurrences of a adjacent. Symbolically, we show that the following pseudo-rewrite rule is admissible:

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.