Use Exercise 4 to construct a multiplication map \(\mu: K(G, n) \times K(G, n) \to K(G, n)\) for any abelian group \(G\), making a CW complex \(K(G, n)\) into an H-space whose multiplication is commutative and associative up to homotopy and has a homotopy inverse. Show also that the H-space multiplication \(\mu\) is unique up to homotopy.
Solution
By Exercise 4, homotopy classes of maps \[ K(G,n)\times K(G,n)\to K(G,n) \] are classified by homomorphisms \[ \pi_n(K(G,n)\times K(G,n))=G\oplus G\to G. \] Choose \(\mu\) corresponding to addition \[ (a,b)\mapsto a+b. \] Its restrictions to the two factors induce the identity on \(G\), hence are homotopic to the identity map. Thus \(\mu\) is an H-space multiplication.
The maps \[ \mu(\mu\times1),\qquad \mu(1\times\mu):K^3\to K \] both induce \((a,b,c)\mapsto a+b+c\) on \(\pi_n\), so Exercise 4 makes them homotopic. Likewise \(\mu\) and \(\mu\tau\) both induce \(a+b\), so multiplication is homotopy-commutative. The map corresponding to \[ a\mapsto-a \] is a homotopy inverse.
Finally any H-space multiplication on \(K(G,n)\) restricts to the identity on both factors, hence induces addition \(G\oplus G\to G\). Exercise 4 therefore implies that it is homotopic to \(\mu\). Thus \[ \boxed{K(G,n)\text{ has a unique H-space multiplication up to homotopy, with the stated properties}.} \]