Introduction

In this chapter we apply contrahomology theory to a particular case of a very general mathematical problem, indicating in 7.5 and 7.7 two directions in which further developments have been made. The general problem concerns a set X with subset A and another set Y; these sets have some structure, that of a group or a difFerentiable manifold or a topological space, for example. Given a transformation g : A → Y, which preserves the structure, the problem is to determine whether g can be extended to a transformation f : X → Y which also preserves the structure.

The case of interest in topology is that in which X, Y are topological spaces and g,f are maps. We shall here be considering the special case in which X, A form a polyhedral pair. If L then is a subcomplex of the complex K (not necessarily finite), we suppose given a map g : |L| → Y and we look for an extension f : |K| → Y. One method of trying to construct f springs early to mind; we can certainly extend g to a map f0 : |K0 ∪ L| → Y, selecting arbitrarily the images of the vertices of K − L, and we can then try to extend f0 to a map f1 : |K1 ∪ L| → Y. If this proves to be impossible, we may profit by our experience of trying to extend f0 and in the light of our difficulties reconsider our selection of f0.