Using an H-space multiplication \(\mu\) on \(K(G, n)\), define an addition in \(\langle X, K(G, n) \rangle\) by \([f] + [g] = [\mu(f, g)]\) and show that under the bijection \(H^n(X; G) \approx \langle X, K(G, n) \rangle\) this addition corresponds to the usual addition in cohomology.
Solution
Let \[ u\in H^n(K(G,n);G) \] be the universal class corresponding to the identity homomorphism of \(G\). For the H-space multiplication \(\mu\), Exercise 6 says that \(\mu_*\) on \(\pi_n=G\oplus G\) is addition. Hence, under the Künneth/representability identification, \[ \mu^*u=\operatorname{pr}_1^*u+\operatorname{pr}_2^*u. \]
If \(f,g:X\to K(G,n)\), then \[ (\mu(f,g))^*u =(f,g)^*\mu^*u =f^*u+g^*u. \] The representing-space bijection \[ \langle X,K(G,n)\rangle\xrightarrow{\cong}H^n(X;G) \] sends \([f]\) to \(f^*u\). Therefore the multiplication-induced addition satisfies \[ \boxed{[f]+[g]\longleftrightarrow f^*u+g^*u,} \] which is exactly the usual addition in cohomology.