Cellular Approximation and CW Approximation

These theorems show that CW complexes are the natural category for homotopy theory: any map can be made cellular up to homotopy, and any space can be replaced by a CW complex without changing its homotopy type.

Cellular Approximation

Theorem9.9Cellular Approximation Theorem

Let $f : X \to Y$ be a continuous map between CW complexes. Then $f$ is homotopic to a cellular map $g : X \to Y$ , meaning $g(X^{(n)}) \subseteq Y^{(n)}$ for all $n$ , where $X^{(n)}$ and $Y^{(n)}$ denote the $n$ -skeleta. Moreover, if $f$ is already cellular on a subcomplex $A \subseteq X$ , the homotopy can be taken to be stationary on $A$ .

The proof is an inductive construction on skeleta, using the fact that any map from a $k$ -cell to a CW complex can be deformed off cells of dimension greater than $k$ .

ExampleMaps between spheres

Consider $f : S^m \to S^n$ with $m < n$ . Give $S^k$ the standard CW structure with one $0$ -cell and one $k$ -cell. By cellular approximation, $f$ is homotopic to a cellular map $g$ with $g(S^m) \subseteq (S^n)^{(m)} = \{*\}$ (since $m < n$ and $S^n$ has no cells in dimensions $1$ through $n-1$ ). Thus $g$ is constant and $f$ is null-homotopic. This proves $\pi_m(S^n) = 0$ for $m < n$ .

CW Approximation

Theorem9.10CW Approximation Theorem

For any topological space $X$ , there exists a CW complex $Z$ and a weak homotopy equivalence $f : Z \to X$ (meaning $f_* : \pi_n(Z) \to \pi_n(X)$ is an isomorphism for all $n$ ). The pair $(Z, f)$ is unique up to homotopy equivalence and is called the CW approximation of $X$ .

Definition

A Postnikov tower for a connected CW complex $X$ is a sequence of fibrations $\cdots \to X_{n+1} \to X_n \to X_{n-1} \to \cdots \to X_1 \to X_0 = *$ together with maps $X \to X_n$ inducing isomorphisms $\pi_k(X) \cong \pi_k(X_n)$ for $k \leq n$ and $\pi_k(X_n) = 0$ for $k > n$ . The fiber of $X_{n+1} \to X_n$ is the Eilenberg-MacLane space $K(\pi_{n+1}(X), n+1)$ .

Eilenberg-MacLane Spaces

ExampleEilenberg-MacLane spaces

An Eilenberg-MacLane space $K(G, n)$ is a CW complex with $\pi_n(K(G,n)) \cong G$ and $\pi_k(K(G,n)) = 0$ for $k \neq n$ . Key examples:

$K(\mathbb{Z}, 1) \simeq S^1$
$K(\mathbb{Z}, 2) \simeq \mathbb{CP}^\infty$
$K(\mathbb{Z}/2, 1) \simeq \mathbb{RP}^\infty$

These spaces represent ordinary cohomology: $H^n(X; G) \cong [X, K(G,n)]$ .

RemarkSignificance for homotopy theory

The CW approximation theorem and Whitehead's theorem together establish that the homotopy category of CW complexes is equivalent to the category of all spaces localized at weak homotopy equivalences. This justifies working exclusively with CW complexes in homotopy theory without loss of generality for homotopy-theoretic questions.