Historical evolution of the concept of homotopic paths

Abstract

The historical evolution of the homotopy concept for paths illustrates how the introduction of a concept (be it implicit or explicit) depends upon the interests of the mathematicians concerned and how it gradually acquires a more satisfactory definition. In our case the equivalence of paths first meant for certain mathematicians that they led to the same value of the integral of a given function or that they led to the same value of a multiple-valued function. (See for instance [Cau], [Pui], [Rie].) Later this dependency upon given functions is dropped (by Jordan for surfaces, by Riemann and most explictly by Poincare) and this leads to a concept which depends only upon the manifold. It thus becomes a concept which belongs to topology. As a consequence of this hesitant evolution, there was at first a confusion between concepts and hence no attention to relations between them. At a later stage these relations were investigated, as for instance the fact that homotopy equivalence implies homology equivalence: in1882, Klein gave an example of a closed curve on a surface which is the boundary of a part of the surface but could not be shrunk to a point. In 1904, Poincare explicitly said that this curve is “homologue a zero” (null homologous), but not “equivalent a zero” (null homotopic). Poincare obtains the homologies from the fundamental group by allowing changes in the order of the terms in the “equivalences”. This means also that the equivalences imply homologies but not vice versa. From a methodological standpoint, this situation of using properties without asking about relations between them or without even properly defining them is reminiscent of mathematics of a century earlier. An example is the way differentiability was used for the differentiation of functions without consciously questioning the properties of this concept until in the nineteenth century examples were given by Bolzano (1834), Weierstrass (1872), Riemann (1854) and others of functions which are continuous but nowhere differentiable [Kli]. Another example is the use eighteenth-century mathematicians made of series without inquiring about the validity of the operations they used on them. Another aspect which had its influence was the success of algebraic methods in topology which explains the preference for theories with “base point” and constrained deformation even though free deformation is a more natural concept. As we can see the evolution of the homotopy concept for paths did not progress without impediments. It also had an influence on the evolution of the abstract group concept and the basic principle of equivalence. These aspects make it a history which is not only intrinsically interesting (how did homotopic paths come to be?), but also because it illustrates relations between different branches of mathematics.