Consider a pushout diagram where \(B\) is \(B\) with \(X\) attached along \(A\) via \(f\). Show that if \(A \hookrightarrow X\) is a cofibration, so is \(B \hookrightarrow B \sqcup_f X\).
Solution
1. Homotopy Extension Property (HEP) setup for the pushout:
1.1. Let \(i: A \hookrightarrow X\) be a closed cofibration and let \(f: A \to B\) be the attaching map.Let \(P = B \sqcup_f X\) be the pushout space, with canonical inclusions \(j: B \hookrightarrow P\) and \(q: X \to P\).
definition of adjunction space / pushout.
1.2. To show \(j: B \hookrightarrow P\) is a cofibration, let \(Y\) be an arbitrary topological space, \(u: P \to Y\) a continuous map, and \(H: B \times I \to Y\) a continuous homotopy such that \(H(b, 0) = u(j(b))\) for all \(b \in B\).
test data for the Homotopy Extension Property.
2. Lifting to \(X\) via the cofibration \(A \hookrightarrow X\):
2.1. Define the initial map on \(X\): \(u_X = u \circ q: X \to Y\).
composition of continuous maps.
2.2. Define the homotopy on \(A\): \(h_A = H \circ (f \times \operatorname{id}_I): A \times I \to Y\).For \(t = 0\): \(h_A(a, 0) = H(f(a), 0) = u(j(f(a))) = u(q(i(a))) = u_X(i(a))\).
pushout identification \(j(f(a)) = q(i(a))\) in \(P\).
2.3. Since \(i: A \hookrightarrow X\) is a cofibration, the pair \((u_X, h_A)\) satisfies the HEP for \((X, A)\).Thus there exists a continuous homotopy \(H_X: X \times I \to Y\) such that:\[ H_X(x, 0) = u_X(x) \quad \text{for all } x \in X, \qquad H_X(i(a), t) = h_A(a, t) = H(f(a), t) \quad \text{for all } (a, t) \in A \times I. \]
definition of cofibration for \(A \hookrightarrow X\).
3. Pasting homotopies on the pushout \(P \times I\):
3.1. Since \(P \times I \cong (B \times I) \sqcup_{f \times \operatorname{id}_I} (X \times I)\) is the pushout of \(B \times I\) and \(X \times I\) along \(A \times I\), define \(\widetilde{H}: P \times I \to Y\) by:\[ \widetilde{H}(p', t) = \begin{cases} H(b, t) & \text{if } p' = j(b) \in j(B), \\ H_X(x, t) & \text{if } p' = q(x) \in q(X). \end{cases} \]
universal property of pushouts.
3.2. On the overlap \(A \times I\), \(H(f(a), t) = H_X(i(a), t)\) by 2 step 3, so \(\widetilde{H}\) is well-defined and continuous on \(P \times I\).
Gluing Lemma for pushouts.
3.3. Verify the initial condition:For \(b \in B\), \(\widetilde{H}(j(b), 0) = H(b, 0) = u(j(b))\).For \(x \in X\), \(\widetilde{H}(q(x), 0) = H_X(x, 0) = u_X(x) = u(q(x))\).Thus \(\widetilde{H}(\cdot, 0) = u\).
evaluation at \(t = 0\).
3.4. Furthermore, \(\widetilde{H}(j(b), t) = H(b, t)\), so \(\widetilde{H}\) extends \(H\).
definition of \(\widetilde{H}\).
4. Conclusion:\(j: B \hookrightarrow B \sqcup_f X\) satisfies the Homotopy Extension Property, so cofibrations are preserved under pushouts. Q.E.D.
1 through 3.