Some problems in topology

Broadly speaking, we may say that analysis situs, or topology, deals with the properties of geometrical figures that remain invariant when the figures are subjected to arbitrary continuous transformations. There are, however, several distinct kinds of analysis situs, because there are several distinct ways of interpreting the physical notion of continuity in mathematical language. For example, there is what we call point theoretical analysis situs which is different in spirit as well as in content from the sort of analysis situs originally proposed by Leibnitz. This branch of the science is essentially an outgrowth of function theory, whereas what Leibnitz had in mind was a new and independent type of mathematics, especially designed to avoid the complications of function theory and to deal directly with the purely qualitative aspects of geometrical problems. No doubt combinatorial analysis situs is more nearly a development of Leibnitz's original idea. The vogue for point theoretical analysis situs seems to be due, in large part, to the predominating influence of analysis on mathematics in general. Nowadays, we tend, almost automatically, to identify physical space with the space of three real variables arid to interpret physical continuity in the classical function theoretical manner. But the space of three real variables is not the only possible mathematical model of physical space, nor is it a perfectly satisfactory model for dealing with certain types of problems. Whenever we attack a topological problem by analytic methods it almost invariably happens that to the intrinsic difficulties of the problem, which we can hardly hope to avoid, there are added certain extraneous difficulties in no way connected with the problem itself, but apparently associated with the particular type of machinery used in dealing with it. Consider, for example, our old friend, the problem of the knotted string. In physical terms, what we have is this. We take an ordinary piece of string, tangle it up in an arbitrary manner, and seal its two ends together. We then ask ourselves whether we have really tied a knot in the string or whether the twists can all be disentangled without breaking the seal that holds the ends together. Now, to solve this problem it is evidently immaterial to know the exact shape of the string. We merely need to know something about the way in which the various branches lace over and under one another, all of which can be described with sufficient accuracy by a rough picture (fig. l a ) , with some device (such as the system of dots shown in the figure), to distinguish the ,,upper" from the ,,lower" branch at each apparent crossing point. Moreover, all the really