Classifying mappings of surfaces
Geoffrey Hemion · 1993
Abstract Let f and g be two given homeomorphisms of a surface S onto itself. We have seen in Chapter 18 that homeomorphisms can be described in terms of lists of numbers. Furthermore the lengths of these lists are no greater than the size of the homeomorphism. The question is, does there exist a further homeomorphism h: S→ S such that f is isotopic to h-1gh? According to the reasoning of the last chapter, we can find a definite (but large) number M(f, g), which depends only on the sizes of the known homeomorphisms f and g, such that if there exists a conjugating homeomorphism, then there also exists one of size no greater than M(f, g). (This is the number referred to as N in Chapter 18.) It is now, in principle at least, a simple matter to go through and check all possible homeomorphisms of size less than this number, looking for some conjugating homeomorphism. Our hypothetical computer will be set the task of going through all lists of lists with no more than M(f, g) entries. Of course it is true that most lists which might be chosen at random will not represent homeomorphisms of the surface. A number of rules will have to be formulated to identify lists which do, in fact, represent homeomorphisms. Furthermore, combinatorial rules will have to be formulated to identify a conjugating homeomorphism during the search. These are routine tasks which we can leave to the hypothetical programmer of our hypothetical computer.