Representability of cohomology by Eilenberg–MacLane spaces

Remarks

One proof uses obstruction theory. A map on the $n$-skeleton that extends over the $(n+1)$-skeleton is given, up to homotopy, by a cellular cocycle. The obstructions to extending it over higher cells lie in groups $H^{k+1}(X; \pi_k K(G, n))$ with $k > n$, which vanish. Another proof checks that $X \mapsto H^n(X; G)$ satisfies the hypotheses of the Brown representability theorem and identifies the representing space.

Example: $K(\Z, 2) = \C P^\infty$, so $H^2(X; \Z)$ is the set of homotopy classes of maps $X \to \C P^\infty$. For $X$ a CW complex, this set also classifies complex line bundles on $X$, and the bijection sends a line bundle to its first Chern class.

The CW hypothesis on $X$ matters: for spaces that are not of the homotopy type of a CW complex, singular cohomology need not be represented in this way. For example, the Warsaw circle has trivial singular cohomology in positive degrees but nontrivial Čech cohomology, and maps into $K(\Z, 1) = S^1$ see the Čech groups.

Going deeper

For a student

This theorem turns cohomology into a homotopy-theoretic object. A natural operation $H^n(-; G) \to H^m(-; G')$ then corresponds, by the Yoneda lemma, to a class in $H^m(K(G, n); G')$, so cohomology operations are computed from the cohomology of Eilenberg–MacLane spaces. The spaces $K(G, n)$ form an $\Omega$-spectrum, since $\Omega K(G, n+1) \simeq K(G, n)$, and this spectrum represents ordinary cohomology in the stable setting.