Show the suspension of an acyclic CW complex is contractible.
Solution
Goal: Show the suspension \(\Sigma X\) is contractible when \(X\) is acyclic.
1. Use reduced homology of suspension.
1.1. For reduced homology of a suspension, \(\widetilde H_n(\Sigma X)\cong \widetilde H_{n-1}(X)\) for \(n\ge2\).
1.2. Since \(X\) is acyclic, \(\widetilde H_{n-1}(X)=0\) for all \(n-1\ge0\), so \(\widetilde H_n(\Sigma X)=0\) for all \(n\ge2\).
1.3. The space \(\Sigma X\) is nonempty and connected, hence \(\widetilde H_0(\Sigma X)=0\).
1.4. Therefore \(\widetilde H_n(\Sigma X)=0\) for every \(n\ge0\).
2. \(\Sigma X\) is simply connected.
2.1. The suspension \(\Sigma X = CX \cup_{X} CX\) is path-connected: any two points lie in one of the two cones, and each cone is path-connected (every point is connected to the cone vertex by a straight line).
2.2. Let \(\gamma \colon S^1 \to \Sigma X\) be a loop. The image \(\gamma(S^1)\) is compact, so the open cover \(\{\mathring{C}_+, \mathring{C}_-\}\) of \(\Sigma X\) by the interiors of the two cones has a finite subcover.
2.3. Break \(\gamma\) into finitely many arcs, each lying in one cone. Since each cone \(C_\pm\) is contractible (it deformation-retracts onto its vertex), any arc in \(C_\pm\) is null-homotopic relative to its endpoints.
2.4. Concatenating these null-homotopies (along the finitely many junction points) gives a null-homotopy of \(\gamma\) in \(\Sigma X\).
2.5. Hence \(\pi_1(\Sigma X) = 1\).
3. Apply Whitehead’s theorem for CW complexes.
3.1. \(\Sigma X\) is a CW complex (suspension of a CW complex).
3.2. A simply connected CW complex with all reduced homology zero is contractible.
3.3. Therefore \(\Sigma X\) is contractible.