Introduction

Whitehead’s Theorem tells us that weak homotopy equivalences between CW complexes are homotopy equivalences. This lecture develops two approximation theorems that make CW complexes easier to work with: every map between CW complexes is homotopic to a cellular map, and every space admits a weak homotopy equivalence from a CW complex.

As a first application, cellular approximation will show that \(\pi_k(S^n) = 0\) for \(k < n\text{.}\) With the usual CW structure on \(S^n\text{,}\) its \(k\)-skeleton is a point whenever \(k < n\text{.}\) A cellular map \(S^k \to S^n\) must therefore be constant. The theorem allows us to replace any based map by such a cellular map through a based homotopy.

We begin with the geometric input: attaching cells of dimension greater than \(n\) gives an \(n\)-connected pair. Applying HELP to the inclusions of skeleta then proves cellular approximation. Finally, we construct a CW approximation of an arbitrary space by attaching cells to make its homotopy groups agree with those of the given space.