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Section 14.4 Exercises

Exercise 14.13. The Hurewicz homomorphism.

Let \((X,x_0)\) be a based space and \(n\geq1\text{.}\) Fix a generator \(u\in\widetilde H_n(S^n;\mathbb Z)\text{.}\) Show that
\begin{equation*} h\colon\pi_n(X,x_0)\longrightarrow H_n(X;\mathbb Z), \qquad [f]\longmapsto f_*(u), \end{equation*}
is well-defined and is a group homomorphism, including when \(n=1\) and \(\pi_1(X,x_0)\) is nonabelian. Hint: For well-definedness, use homotopy invariance of homology. For the homomorphism property, represent the product by the pinch map \(p\colon S^n\to S^n\vee S^n\) followed by the two representative maps. Show that \(p_*(u)=(u,u)\) under the usual identification of the wedge’s homology with \(\mathbb Z\oplus\mathbb Z\text{.}\)

Exercise 14.14. Uniqueness from representability.

Let \(G\) be abelian, let \(n\geq1\text{,}\) and let \(K\) and \(L\) be CW complexes of type \(K(G,n)\text{,}\) with chosen identifications of their \(n\)th homotopy groups with \(G\text{.}\) Use Corollary 14.12 to construct maps \(f\colon K\to L\) and \(g\colon L\to K\) inducing the identity on \(G\text{.}\) Show that \(g\circ f\simeq\mathrm{id}_K\) and \(f\circ g\simeq\mathrm{id}_L\text{.}\) Deduce a second proof of uniqueness for abelian \(G\text{,}\) including all cases with \(n\geq2\text{.}\) Hint: What homomorphisms correspond to the composites and the identity maps?

Exercise 14.15. Recognizing a sphere.

Let \(n\geq2\) and let \(X\) be a simply connected CW complex with
\begin{equation*} \widetilde H_i(X;\mathbb Z)\cong \begin{cases} \mathbb Z & i=n,\\ 0 & i\neq n. \end{cases} \end{equation*}
Show that \(X\simeq S^n\text{.}\) Hint: Use Hurewicz inductively to show that \(X\) is \((n-1)\)-connected. Realize a generator of \(H_n(X)\) by a map \(S^n\to X\text{,}\) then apply Corollary 14.4.

Exercise 14.16. Mapping cones.

Let \(f\colon X\to Y\) be a map between simply connected CW complexes. Show that if its mapping cone \(C_f=Y\cup_f CX\) is contractible, then \(f\) is a homotopy equivalence. Hint: Use the reduced homology exact sequence of the mapping cone and Corollary 14.4.
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