Homotopy Groups and Bundles Over Spheres

Abstract

We saw in Chapter 2 that the concept of homotopy plays a central role in bundle theory. Although we have so far dealt only with differentiable maps, it is more convenient, when working with homotopies, to consider continuous maps, and we will do so in this chapter. One purpose of this section is to try and convince the reader that we are not introducing new objects when, for example, we consider the pullback f*ξ of a bundle via a continuous map f: Explicitly, we will show that any continuous map between manifolds is homotopic to a differentiable one, and the latter can be chosen to be arbitrarily close to the original one.