Section 9.2 The CW Approximation Theorem

We now show that, up to weak homotopy equivalence, every space is a CW complex. The basic construction is to attach cells to kill unwanted homotopy classes without changing the homotopy groups in lower degrees.

Lemma 9.14.

Let \(X\) be any space and \(k\geq1\text{.}\) A map \(j\colon S^k\to X\) is null-homotopic if and only if it extends to a map \(D^{k+1}\to X\text{.}\) If \(j\) is based, these conditions are also equivalent to \([j]=0\) in \(\pi_k(X,j(s_0))\text{.}\)

Proof.

A null-homotopy of \(j\) descends to a map from the cone \(CS^k\cong D^{k+1}\text{,}\) extending \(j\) on its boundary. Conversely, given an extension \(F\colon D^{k+1}\to X\text{,}\) contract the disc linearly to a chosen point \(s_0\in S^k\text{.}\) The formula

\begin{equation*} H(s,t)=F\bigl((1-t)s+ts_0\bigr) \end{equation*}

defines a null-homotopy of \(j\) which fixes \(s_0\text{.}\) Thus, for a based map, the extension also gives a based null-homotopy, proving the last assertion.

Lemma 9.15.

Let \(X\) be any space and \(n\geq0\text{.}\) There is a space \(Y\) and an inclusion \(i\colon X\hookrightarrow Y\) inducing isomorphisms on \(\pi_q\) for \(0\leq q\leq n\text{,}\) with \(\pi_{n+1}(Y,i(x))=0\) for every \(x\in X\text{.}\)

Proof.

Choose a basepoint in each path component of \(X\text{,}\) and let \(J\) index representatives \(j\colon S^{n+1}\to X\) of all elements of \(\pi_{n+1}\) at these basepoints. Attach a cell along each representative:

By Lemma 9.2, \((Y,X)\) is \((n+1)\)-connected. The long exact sequence, as explained after Definition 8.2, therefore makes \(i\) an \((n+1)\)-equivalence: it induces isomorphisms through degree \(n\) and surjections in degree \(n+1\text{.}\) Every class in \(\pi_{n+1}(Y)\) therefore comes from a class in \(\pi_{n+1}(X)\text{.}\) For each representative \(j\text{,}\) the composite \(i\circ j\colon S^{n+1}\to Y\) extends over its attached disc \(D^{n+2}\text{.}\) By Lemma 9.14, \(i\circ j\) is based null-homotopic, so \(i_*[j]=0\text{.}\) Since \(i_*\) is surjective, \(\pi_{n+1}(Y)=0\text{.}\) Change of basepoint gives the assertion at every point of \(X\text{.}\)

Definition 9.16.

A CW approximation of a space \(X\) is a CW complex \(\Gamma X\) together with a weak homotopy equivalence \(\gamma\colon\Gamma X\to X\text{.}\)

Theorem 9.17. CW approximation.

Every space \(X\) admits a CW approximation \(\gamma\colon\Gamma X\to X\text{.}\)

Proof.

We can assume that \(X\) is path-connected, or we can work one path component at a time. Choose a basepoint and a set of based representatives \(J = \{ j_q \mid [j] \in \pi_qX, q \ge 1 \}\text{.}\) Let \(\Gamma_1 X = \bigvee_{j_q \in J} S^q\) (with its standard CW structure) and \(\gamma_1 \coloneqq \vee j_q \colon \Gamma_1 X \to X\text{.}\) By construction, \(\gamma_1\) is surjective on \(\pi_q\) for every \(q\geq1\text{.}\) Since \(\Gamma_1 X\) and \(X\) are path-connected, it also induces a bijection on \(\pi_0\text{,}\) and hence is a \(1\)-equivalence. Suppose by induction that we have constructed a CW complex \(\Gamma_n X\) and an \(n\)-equivalence \(\Gamma_n X \to X\text{.}\) Once again, let \(J = \{ j \mid [j] \in \pi_n(\Gamma_n X), [\gamma_n \circ j] = 0 \in \pi_nX \}\text{.}\) Choose the representatives \(j\) to be cellular, so that attaching the new cells preserves the CW structure. As in the proof of Lemma 9.15, we attach cells to kill these classes; here we kill only the kernel. By Lemma 9.14, for each \(j \in J\) there exists an extension \(h_j\) of \(S^n \xrightarrow{\gamma_n \circ j} X\) to \(D^{n+1}\text{.}\) We then construct \(\Gamma_{n+1}X\) by the pushout

The inclusion \(i_n\colon \Gamma_n X\hookrightarrow \Gamma_{n+1}X\) is an \(n\)-equivalence by Lemma 9.2, just as in Lemma 9.15. Thus every based map \(S^q\to \Gamma_{n+1}X\) for \(q\leq n\) factors through \(\Gamma_n X\) up to based homotopy. Since \(\gamma_{n+1}i_n=\gamma_n\) and \((i_n)_*\) is an isomorphism for \(q<n\text{,}\) \((\gamma_{n+1})_*\) is also an isomorphism in these degrees. For \(q=n\) we have a commutative diagram

so that \((\gamma_{n+1})_*\) is surjective. For injectivity, let \(u\in\pi_n(\Gamma_{n+1}X)\) map to zero and lift it to \(v\in\pi_n(\Gamma_n X)\text{.}\) Then \((\gamma_n)_*v=0\text{,}\) so \(v\) is represented by one of the attaching maps. Its attached disc makes \((i_n)_*v=0\text{,}\) hence \(u=0\text{.}\) Thus \((\gamma_{n+1})_*\) is an isomorphism in degree \(n\text{.}\)

Moreover, \(\gamma_1\) factors through \(\gamma_{n+1}\) via \(\Gamma_1 X\hookrightarrow \Gamma_{n+1}X\text{.}\) Since \(\gamma_1\) is surjective on every homotopy group, so is \(\gamma_{n+1}\text{,}\) in particular in degree \(n+1\text{.}\) Hence \(\gamma_{n+1}\) is an \((n+1)\)-equivalence, completing the induction.

Finally, set \(\Gamma X=\operatorname{colim}_n \Gamma_n X\) with its CW topology and \(\gamma=\operatorname{colim}_n\gamma_n\text{.}\) All cells added after \(\Gamma_{q+1}X\) have dimension at least \(q+2\text{,}\) so \(\pi_q(\Gamma X)\cong\pi_q(\Gamma_{q+1}X)\cong\pi_q(X)\) for \(q\geq1\text{.}\) The spaces are path-connected, and therefore \(\gamma\) is a weak homotopy equivalence. For general \(X\text{,}\) take the coproduct of the constructions in its path components.

Remark 9.18.

There is also a CW approximation for pairs: for any pair \((X,A)\) there is a CW pair \((\Gamma X,\Gamma A)\) and a map of pairs to \((X,A)\) whose maps \(\Gamma X\to X\) and \(\Gamma A\to A\) are weak homotopy equivalences. In particular, \(\Gamma A\) is a subcomplex of \(\Gamma X\text{.}\)

Exercise 9.19.

Let \(f\colon X\to Y\) be a map, and choose CW approximations \(\gamma_X\colon\Gamma X\to X\) and \(\gamma_Y\colon\Gamma Y\to Y\text{.}\) Show that there is a map \(\Gamma f\colon\Gamma X\to\Gamma Y\text{,}\) unique up to homotopy, such that the square

commutes up to homotopy. Deduce that any two CW approximations of \(X\) are homotopy equivalent by an equivalence commuting with their maps to \(X\) up to homotopy. Hint: Apply Theorem 8.10.

Exercise 9.20.

Suppose given maps \(f\colon X\to Y\) and \(g\colon Y\to X\) with \(g\circ f\simeq\mathrm{id}_X\text{.}\) We say that \(Y\) dominates \(X\text{.}\) If \(Y\) is a CW complex, prove that \(X\) has the homotopy type of a CW complex.

Exercise 9.21. Extension Lemma.

Let \((X,A)\) be a CW pair and \(Y\) a nonempty path-connected space. Let \(f\colon A\to Y\) be a map. Suppose that \(\pi_{n-1}(Y)=0\) for every \(n\geq2\) for which \(X\) has an \(n\)-cell outside \(A\text{.}\) Show that \(f\) extends to a map \(X\to Y\text{.}\) Hint: Extend over the relative cells using Lemma 9.14. Nonemptiness and path-connectedness handle the relative \(0\)- and \(1\)-cells, respectively.

Exercise 9.22.

Show that a CW complex retracts onto any nonempty contractible subcomplex. Hint: Apply Exercise 9.21 to the identity map of the subcomplex.

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