CW Complexes and Homotopy

CW complexes are topological spaces equipped with a partitioning into compact pieces called “cells.” They are particularly suitable for group theory: a presentation of a group can be interpreted as a recipe for building a twodimensional CW complex (Example 1.2.17), and we will see in later chapters that CW complexes exhibit many group theoretic properties geometrically.

Beginners in algebraic topology are usually introduced first to simplicial complexes. A simplicial complex is (or can be interpreted as) an especially nice kind of CW complex. In the long run, however, it is often unnatural to be confined to the world of simplicial complexes, in particular because they often have an inconveniently large number of cells. For this reason, we concentrate on CW complexes from the start. Simplicial complexes are treated in Chap. 5.