Lemma 7.20.
If \(\{(X_\alpha, A_\alpha)\}\) is a collection of pairs each satisfying the HEP, then so does \(\bigl(\coprod_\alpha X_\alpha,\, \coprod_\alpha A_\alpha\bigr)\text{.}\)
Section 7.3 CW Pairs Satisfy the HEP
Following the principle of Remark 7.9, we now show that the HEP is preserved by the three constructions used to build CW complexes from discs. In each proof we use the fact that taking the product with \(I\) preserves these constructions: \((\coprod X_\alpha) \times I = \coprod (X_\alpha \times I)\text{,}\) the product \((X \cup_A B) \times I\) is the pushout of \(X \times I \leftarrow A \times I \to B \times I\text{,}\) and \((\bigcup_n X_n) \times I\) has the weak topology with respect to the \(X_n \times I\text{.}\) All of these hold because \(I\) is compact Hausdorff, which implies that the product of a quotient map with \(\mathrm{id}_I\) is again a quotient map.
Proof.
Exercise.
Lemma 7.21.
If \((X, A)\) satisfies the HEP and \(f \colon A \to B\) is a continuous map, then the pushout pair \((X \cup_A B,\, B)\) satisfies the HEP, where \(X \cup_A B\) is the pushout of \(X \hookleftarrow A \xrightarrow{f} B\text{.}\)
Proof.
Write \(Z = X \cup_A B\) and let \(\varphi \colon X \to Z\) and \(\psi \colon B \to Z\) be the canonical maps, so that \(\varphi|_A = \psi \circ f\text{.}\) Suppose given \(g \colon Z \to Y\) and a homotopy \(G \colon B \times I \to Y\) with \(G_0 = g \circ \psi\text{.}\) Then \(g \circ \varphi \colon X \to Y\) is a map whose restriction to \(A\) is \(g \circ \psi \circ f = G_0 \circ f\text{,}\) so \(G \circ (f \times \mathrm{id}_I) \colon A \times I \to Y\) is a homotopy starting at \((g \circ \varphi)|_A\text{.}\) By the HEP for \((X, A)\) it extends to a homotopy \(\widetilde{G} \colon X \times I \to Y\) starting at \(g \circ \varphi\text{.}\) Now \(\widetilde{G}\) and \(G\) agree on \(A \times I\) (after composing with \(f \times \mathrm{id}_I\)), so they glue to a map \(Z \times I \to Y\) by the universal property of the pushout \(Z \times I = (X \times I) \cup_{A \times I} (B \times I)\text{.}\) This is the required extension: it restricts to \(G\) on \(B \times I\) and to \(g\) at time \(0\text{.}\)
Lemma 7.22.
Let \(A = X_{-1} \subseteq X_0 \subseteq \cdots \subseteq X = \bigcup_n X_n\text{,}\) where \(X\) has the weak topology with respect to the \(X_n\text{.}\) If each pair \((X_n, X_{n-1})\) satisfies the HEP, then so does \((X, A)\text{.}\)
Proof.
Suppose given \(g \colon X \to Y\) and a homotopy \(G_{-1} \colon A \times I \to Y\) starting at \(g|_A\text{.}\) Inductively, having constructed \(G_{n-1} \colon X_{n-1} \times I \to Y\) starting at \(g|_{X_{n-1}}\text{,}\) use the HEP for \((X_n, X_{n-1})\) to extend \(G_{n-1}\) to a homotopy \(G_n \colon X_n \times I \to Y\) starting at \(g|_{X_n}\text{.}\) The \(G_n\) are compatible, so they define a function \(\widetilde{G} \colon X \times I \to Y\text{,}\) which is continuous because \(X \times I\) has the weak topology with respect to the \(X_n \times I\text{.}\) By construction \(\widetilde{G}\) restricts to \(G_{-1}\) on \(A \times I\) and to \(g\) at time \(0\text{.}\)
Remark 7.23.
Categorically, Lemma 7.20, Lemma 7.21, Lemma 7.22 express the stability of cofibrations under coproducts, pushouts, and sequential colimits. Together with the fact that cofibrations are closed under composition (a two-step case of Lemma 7.22) and under retracts, this structure can be axiomatised in the framework of model categories, to which we return in Chapter 12.
Theorem 7.24.
Every relative CW complex \((X, A)\) satisfies the HEP. In particular, every CW pair satisfies the HEP.
Proof.
By Proposition 7.16, the pair \((D^n, S^{n-1})\) satisfies the HEP. By Lemma 7.20, so does \(\bigl(\coprod_\alpha D^n,\, \coprod_\alpha S^{n-1}\bigr)\text{.}\) Applying Lemma 7.21 to the defining pushout square
shows that each skeletal pair \((X_n, X_{n-1})\) satisfies the HEP. By Lemma 7.22, the pair \((X, A)\) satisfies the HEP.
Example 7.25.
For a CW complex \(X\text{,}\) the pairs \((X, X_n)\) satisfy the HEP for every \(n\text{.}\) This lets us extend a homotopy on a skeleton to all of \(X\text{.}\) In Chapter 8 we will use the same method of extending over cells to prove HELP, which in turn gives cellular approximation in Chapter 9.
Exercise 7.26.
Let \(f \colon X \to Y\) be a map. Its mapping cylinder is
\begin{equation*}
M_f := \bigl(X \times I \,\sqcup\, Y\bigr) \,\big/\, (x, 1) \sim f(x),
\end{equation*}
which contains \(X = X \times \{0\}\) and \(Y\) as disjoint subspaces. Show that the pair \((M_f, X)\) satisfies the HEP. (Hint: \(M_f\) is the pushout of \(X \times I \hookleftarrow X \times \partial I \xrightarrow{\mathrm{id} \sqcup f} X \sqcup Y\text{,}\) so Exercise 7.15, Lemma 7.21 show that \((M_f, X \sqcup Y)\) has the HEP. Given a homotopy on \(X\text{,}\) extend it to \(X \sqcup Y\) by the constant homotopy on \(Y\text{.}\)) The mapping cylinder will play a central role in Chapter 8.
Exercise 7.27.
Show that if \((X, A)\) and \((Y, X)\) both satisfy the HEP, then so does \((Y, A)\text{.}\) Show also that if \((X, A)\) satisfies the HEP, then so does \((X \times Z, A \times Z)\) for every space \(Z\text{.}\)
