Given abelian groups \(G\) and \(H\) and CW complexes \(K(G, n)\) and \(K(H, n)\), show that the map \(\langle K(G, n), K(H, n) \rangle \to \operatorname{Hom}(G, H)\) sending a homotopy class \([f]\) to the induced homomorphism \(f_*: \pi_n(K(G, n)) \to \pi_n(K(H, n))\) is a bijection.
Solution
Representability gives \[ \langle K(G,n),K(H,n)\rangle \cong H^n(K(G,n);H). \] Since \(K(G,n)\) is \((n-1)\)-connected, Hurewicz gives \[ H_n(K(G,n);\mathbb Z)\cong\pi_n(K(G,n))\cong G, \] and \(H_{n-1}=0\). The universal coefficient theorem therefore gives \[ H^n(K(G,n);H) \cong\operatorname{Hom}(H_n(K(G,n)),H) \cong\operatorname{Hom}(G,H). \] Under these identifications, the class represented by \[ f:K(G,n)\to K(H,n) \] is exactly the homomorphism induced on \(\pi_n\), by naturality of Hurewicz and of the representing cohomology class. Thus \[ \boxed{\langle K(G,n),K(H,n)\rangle\xrightarrow{\cong}\operatorname{Hom}(G,H),\quad[f]\mapsto f_*.} \]