The Group of Mapping Classes

In combinatorial topology, topological concepts are represented by arithmetic concepts. Thus, in principle, all problems of combinatorial topology are reduced to arithmetic problems. However, this reduction is of no use for the resolution of the problems in most cases, because the corresponding arithmetic problems have little relation to known results or methods. This is especially true of all problems in which homotopic transformations are considered non-trivial or, as one can also say, in which the exceedingly numerous and hard to visualize constructions of simply connected polyhedra of different dimensions come into play. In an earlier work* I have attempted to represent this construction arithmetically in an understandable way for two-dimensional polyhedra. In doing so, I showed that homotopy problems yield arithmetic problems which fall outside the extensive domain of group theory. They concern more general operations which are difficult to investigate, the totality of which I have covered by the name “games”.