CW-Complexes

Abstract

Computing homotopy groups is not easy in general, and one very quickly exhausts the supply of useful theorems about the homotopy groups of general topological spaces. One of the difficulties is that given two arbitrary topological spaces X, Y it is very difficult to construct any map f: X → Y. If we restricted our attention to a class of spaces built up step by step out of simple building blocks (think of simplicial complexes, for example), then we might hope to construct maps step by step, extending them over the building blocks one at a time. In this chapter we describe a useful category of such spaces (CW-complexes) and display some of their elementary properties. In the next chapter we shall prove some much deeper homotopy properties of CW-complexes.