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Section 14.3 Eilenberg–MacLane Spaces as Representing Spaces

We now show that cohomology classes can be described by maps into Eilenberg–MacLane spaces. Throughout this section, \(G\) is abelian and each \(K(G,n)\text{,}\) \(n\geq1\text{,}\) is a based CW complex with a chosen identification \(\pi_n(K(G,n))\cong G\text{.}\)
The first step is to find a class on \(K(G,n)\) that corresponds to the identity of \(G\text{.}\) The universal coefficient theorem and Hurewicz give
\begin{align*} H^n(K(G, n); G) &\;\cong\; \mathrm{Hom}_{\mathbb{Z}}\bigl(H_n(K(G, n); \mathbb{Z}),\, G\bigr) && \text{(universal coefficient)}\\ &\;\cong\; \mathrm{Hom}_{\mathbb{Z}}\bigl(\pi_n(K(G, n)),\, G\bigr) && \text{(Hurewicz)}\\ &\;\cong\; \mathrm{Hom}_{\mathbb{Z}}(G, G). \end{align*}
The universal coefficient \(\mathrm{Ext}\) term vanishes: \(H_{n-1}(K(G,n))=0\) for \(n\geq2\text{,}\) while \(H_0(K(G,1))\cong\mathbb Z\) is free. For \(n=1\text{,}\) Hurewicz identifies \(H_1(K(G,1))\) with \(G\) because \(G\) is abelian.
There is therefore a distinguished class \(\iota \in H^n(K(G, n); G)\) — the fundamental class — corresponding to \(\mathrm{id}_G\) under these isomorphisms.
Pulling back \(\iota\) along a map \(f\colon X\to K(G,n)\) gives a class in \(H^n(X;G)\text{.}\) Since pullback depends only on the homotopy class of \(f\text{,}\) we obtain a well-defined map
\begin{equation*} T \colon [X, K(G, n)] \to H^n(X; G), \qquad [f] \mapsto f^*(\iota). \end{equation*}
The theorem says that every cohomology class arises in this way, and that the class determines the map up to homotopy. The proof has three parts. We first give the sets of based homotopy classes natural abelian group structures and verify the cohomology axioms. We then invoke the uniqueness theorem for ordinary cohomology, an external input that identifies this theory from its coefficient groups. Finally, we check that the resulting isomorphism is the specific map \(T\text{,}\) rather than merely some natural isomorphism. A direct proof using cells is also possible (see Hatcher, Theorem 4.57, or [RW, Thm. 1.14.4]).

Proof.
Sketch of Lemma 14.10. Recall the based loop space \(\Omega X = \{\,f \colon I \to X : f(0) = f(1) = x_0\,\}\text{,}\) with bijection
\begin{equation*} [\Sigma X, Y]_* \;\cong\; [X, \Omega Y]_*, \qquad f \mapsto \bigl(x \mapsto f(x, -)\bigr). \end{equation*}
In particular, \(\pi_i(\Omega Y)\cong\pi_{i+1}(Y)\) for \(i\geq1\text{,}\) and \(\pi_0(\Omega Y)\) is identified with \(\pi_1(Y)\text{.}\) Hence \(\Omega K(G,k+1)\) has the homotopy groups of \(K(G,k)\) for \(k\geq0\text{.}\) Choose based weak homotopy equivalences
\begin{equation*} e_k\colon K(G,k)\longrightarrow\Omega K(G,k+1) \end{equation*}
inducing the identity on \(G\) under the chosen identifications. Such maps exist by CW approximation and the uniqueness construction of Lecture 13; for \(k=0\) choose a representative loop for each element of \(G\text{.}\) For \(k<0\text{,}\) the loop spaces in question are points. These weak equivalences induce bijections on based homotopy classes of maps from CW complexes, by the based version of Whitehead’s theorem (Theorem 8.10).
Transport the loop-concatenation group structure on \([X,\Omega K(G,k+1)]_*\) across the bijection induced by \(e_k\text{.}\) This defines a group structure on \(h^k(X)\text{,}\) natural in \(X\text{.}\) Postcomposition with any looped map preserves concatenation. In particular, \(\Omega e_{k+1}\) gives a group isomorphism from \([X,\Omega K(G,k+1)]_*\) to \([X,\Omega^2K(G,k+2)]_*\text{.}\) On the latter set, concatenation in the two loop coordinates satisfies the interchange law and has the same identity. Eckmann–Hilton therefore shows that the transported group structure is abelian. In degree \(0\text{,}\) the identification \(\pi_0(\Omega K(G,1))\cong G\) shows that this operation is pointwise addition; in negative degrees the group is zero.
We must also check that the suspension bijections respect these operations. On \([\Sigma X,K(G,k+1)]_*\) there are two group operations: the one just constructed using \(e_{k+1}\text{,}\) and the one induced by pinching the suspension. Under adjunction and postcomposition with \(\Omega e_{k+1}\text{,}\) these correspond to concatenation in the two coordinates of \([X,\Omega^2K(G,k+2)]_*\text{.}\) Eckmann–Hilton identifies them. Hence the suspension bijections are group isomorphisms, not just bijections of sets. We have obtained functors
\begin{equation*} h^k\colon\mathrm{Ho}(\mathbf{CW}_*)^{\mathrm{op}}\longrightarrow\mathbf{Ab} \end{equation*}
with compatible suspension isomorphisms.
We verify the axioms of a reduced cohomology theory.
  • Suspension. \(h^{n+1}(\Sigma X) = [\Sigma X, K(G, n+1)]_* \cong [X, \Omega K(G, n+1)]_* \cong [X, K(G, n)]_* = h^n(X)\text{.}\)
  • Weak equivalence. A based weak equivalence \(f\colon X\to Y\) of CW complexes is a based homotopy equivalence by Whitehead’s Theorem. Hence \(f^*\colon h^n(Y)\to h^n(X)\) is an isomorphism.
  • Additivity. \(\bigl[\bigvee_i X_i, K(G, n)\bigr]_* \cong \prod_i [X_i, K(G, n)]_*\text{.}\)
  • Exactness. For a based CW pair \((X,A)\text{,}\) a map \(X\to K(G,k)\) whose restriction to \(A\) is based null-homotopic can, by the homotopy extension property, be homotoped to be constant on \(A\text{.}\) It then factors through \(X/A\text{.}\) Thus
    \begin{equation*} h^k(X/A)\longrightarrow h^k(X)\longrightarrow h^k(A) \end{equation*}
    is exact. Applying this to the cofibration sequence \(A\to X\to X/A\to\Sigma A\to\cdots\) and using suspension gives the long exact sequence (see Hatcher pp. 397–399).
The preceding lemma supplies the cohomology theory. To identify it with ordinary cohomology, we use the following uniqueness theorem without proof (Hatcher, Theorem 4.59). Its normalization on coefficients will let us identify the universal class.

Proof.
Proof of Theorem 14.9. The coefficient groups of \(h^*\) are
\begin{equation*} h^k(S^0)\cong \begin{cases} G & k=0,\\ 0 & k\neq0. \end{cases} \end{equation*}
Thus Lemma 14.11 gives natural isomorphisms
\begin{equation*} U_X\colon [X,K(G,n)]_*\xrightarrow{\;\cong\;}\widetilde H^n(X;G) \end{equation*}
normalized to induce the identity on coefficients. In particular, for \(a\colon S^n\to K(G,n)\text{,}\) the class \(U_{S^n}([a])\) corresponds to \([a]\in G\) under \(\widetilde H^n(S^n;G)\cong G\text{.}\)
Let \(K=K(G,n)\) and put \(\eta=U_K([\mathrm{id}_K])\text{.}\) Naturality gives
\begin{equation*} U_X([f])=f^*\eta. \end{equation*}
For every based \(a\colon S^n\to K\text{,}\) we have \(a^*\eta=U_{S^n}([a])\text{,}\) which evaluates on the fundamental class of \(S^n\) as \([a]\in G\text{.}\) Under the universal coefficient and Hurewicz identifications above, this says that \(\eta\) corresponds to \(\mathrm{id}_G\text{.}\) Hence \(\eta=\iota\text{,}\) so \(U_X([f])=f^*\iota\text{.}\)
Finally, for an arbitrary CW complex \(X\text{,}\) let \(X_+=X\sqcup\{*\}\) with the added point as basepoint. There are natural identifications
\begin{equation*} [X_+,K(G,n)]_*\cong[X,K(G,n)],\qquad \widetilde H^n(X_+;G)\cong H^n(X;G). \end{equation*}
Applying \(U\) to \(X_+\) gives the asserted isomorphism \(T\text{.}\) The group structure on unbased homotopy classes is obtained through this identification.
As a first application, we can classify maps between Eilenberg–MacLane spaces.

Proof.
We have bijections
\begin{align*} &\cong H^n\bigl(K(G, n); G'\bigr)\\ &\cong \Hom\bigl(H_n(K(G, n)), G'\bigr)\\ &\cong \Hom(G, G'). \end{align*}
The first is Theorem 14.9. The second is the universal coefficient theorem, since the term \(\mathrm{Ext}(H_{n-1}(K(G,n)), G')\) vanishes (\(H_{n-1}\) is \(0\) for \(n \geq 2\) and free for \(n = 1\)). The third is Hurewicz’s theorem (Theorem 14.2, Theorem 14.8), using that \(G\) is abelian when \(n = 1\text{.}\)
Under this bijection, a map corresponds to its induced homomorphism on \(\pi_n\text{,}\) by naturality of the universal coefficient and Hurewicz maps. In particular, composition of maps corresponds to composition of homomorphisms. This gives another route to the uniqueness theorem of Lecture 13, explored in Exercise 14.14.
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