Use the Extension Lemma (homotopy extension property for CW pairs) to show that a CW complex \(X\) retracts onto any contractible subcomplex \(A \subseteq X\).
Solution
Goal: Prove that if \((X, A)\) is a CW pair with \(A\) contractible, there exists a continuous retraction \(r: X \to A\) (\(r|_A = \operatorname{id}_A\)).
1. The Homotopy Extension Property (HEP) / Extension Lemma for CW Pairs:Proof:
1.1. A subcomplex \(A\) of a CW complex \(X\) forms a CW pair \((X, A)\).
1.2. By the Homotopy Extension Property (HEP) (Hatcher, Lemma 4.15 / Extension Lemma):For any CW pair \((X, A)\), the pair has the homotopy extension property with respect to every topological space \(Y\).
1.3. Specifically: Given a continuous map \(f: X \to Y\) and a homotopy \(h: A \times I \to Y\) starting at \(f|_A\) (i.e. \(h(a, 0) = f(a)\) for all \(a \in A\)), there exists a continuous homotopy \(H: X \times I \to Y\) such that:- \(H(x, 0) = f(x)\) for all \(x \in X\), and- \(H(a, t) = h(a, t)\) for all \((a, t) \in A \times I\).
2. Setup for Contractible Subcomplex \(A\):Proof:
2.1. Let \(A \subseteq X\) be a contractible subcomplex.
2.2. By definition of contractibility, the identity map \(\operatorname{id}_A: A \to A\) is homotopic to a constant map \(c_{a_0}: A \to A\) (where \(c_{a_0}(a) = a_0 \in A\)).
2.3. That is, there exists a continuous homotopy:\[h: A \times [0, 1] \longrightarrow A\]such that \(h(a, 0) = a = \operatorname{id}_A(a)\) and \(h(a, 1) = a_0\) for all \(a \in A\).
2.4. Reversing the time parameter \(t \mapsto 1 - t\), we get a continuous homotopy:\[\bar{h}: A \times [0, 1] \longrightarrow A, \qquad \bar{h}(a, t) = h(a, 1 - t)\]satisfying \(\bar{h}(a, 0) = a_0\) (constant map) and \(\bar{h}(a, 1) = a = \operatorname{id}_A(a)\).
3. Extending the Homotopy to \(X\):Proof:
3.1. Set the target space to be \(Y = A\).
3.2. Define the initial map \(f_0: X \to A\) to be the constant map:\[f_0(x) = a_0 \quad \text{for all } x \in X.\]
3.3. Notice that the restriction of \(f_0\) to \(A\) matches the start of the homotopy \(\bar{h}\):\[f_0|_A(a) = a_0 = \bar{h}(a, 0) \quad \text{for all } a \in A.\]
3.4. Since \((X, A)\) is a CW pair, by the Homotopy Extension Property (with target space \(Y = A\)), there exists a continuous homotopy:\[H: X \times [0, 1] \longrightarrow A\]such that:- \(H(x, 0) = f_0(x) = a_0\) for all \(x \in X\), and- \(H(a, t) = \bar{h}(a, t)\) for all \(a \in A, t \in [0, 1]\).
4. Constructing the Retraction \(r: X \to A\):Proof:
4.1. Define the map \(r: X \to A\) by evaluating \(H\) at time \(t = 1\):\[r(x) \coloneqq H(x, 1).\]
4.2. Since \(H\) is continuous and has image in \(A\), \(r: X \to A\) is a continuous map.
4.3. For any point \(a \in A\):\[r(a) = H(a, 1) = \bar{h}(a, 1) = \operatorname{id}_A(a) = a.\]
4.4. Thus \(r|_A = \operatorname{id}_A\), which proves that \(r: X \to A\) is a retraction.
5. Conclusion:\(X\) retracts onto the contractible subcomplex \(A\). Q.E.D.