Section 13.3 Uniqueness

We have constructed one \(K(G, n)\text{.}\) We now show that any two such spaces are homotopy equivalent.

Lemma 13.12.

Let \(X = \bigvee_\alpha S^n_\alpha \cup \bigcup_\beta e^{n+1}_\beta\) for some \(n \geq 1\text{,}\) and let \(Y\) be a path-connected space. For every homomorphism \(\psi \colon \pi_n(X) \to \pi_n(Y)\text{,}\) there is a based map \(f \colon X \to Y\) inducing \(\psi\) on \(\pi_n\text{.}\)

Proof.

Define \(f\) on the \(0\)-skeleton by sending the wedge point to the basepoint of \(Y\text{.}\) For each generator \(\alpha\text{,}\) let \(i_\alpha \colon S^n_\alpha \hookrightarrow X\) be the inclusion and extend \(f\) over \(S^n_\alpha\) by a representative of \(\psi([i_\alpha]) \in \pi_n(Y)\text{.}\) This defines \(f\) on the wedge \(\bigvee_\alpha S^n_\alpha\text{.}\)

Write \(W = \bigvee_\alpha S^n_\alpha\) and \(j \colon W \hookrightarrow X\text{.}\) The homomorphisms \(f_*\) and \(\psi \circ j_*\) from \(\pi_n(W)\) to \(\pi_n(Y)\) agree on the generators \([i_\alpha]\text{,}\) so they are equal. To extend over a cell \(e^{n+1}_\beta\) with attaching map \(\varphi_\beta \colon S^n \to W\text{,}\) we need \(f \circ \varphi_\beta\) to be null-homotopic. But

\begin{equation*} [f \circ \varphi_\beta] \;=\; f_*[\varphi_\beta] \;=\; \psi(j_*[\varphi_\beta]) \;=\; 0, \end{equation*}

since \(\varphi_\beta\) extends over the cell, so \(j_*[\varphi_\beta] = 0\) in \(\pi_n(X)\text{.}\) Hence \(f\) extends over \(X\text{.}\) The extension induces \(\psi\) on \(\pi_n\text{,}\) because it agrees with \(\psi\) after composing with \(j_*\text{,}\) and \(j_*\) is surjective since \((X, W)\) is \(n\)-connected (Lemma 9.2).

Proposition 13.13.

The homotopy type of a CW complex \(K(G, n)\) is uniquely determined by \(G\) and \(n\text{.}\)

Proof.

It suffices to show that the CW complex \(K\) constructed in the proof of Theorem 13.10 is homotopy equivalent to every CW complex \(K'\) of type \(K(G, n)\text{.}\) In particular, \(K_{n+1} = X = \bigvee_\alpha S^n_\alpha \cup \bigcup_\beta e^{n+1}_\beta\text{.}\) By Lemma 13.12, the chosen isomorphism \(\pi_n(X) \cong \pi_n(K) \cong \pi_n(K')\) is realized by a map \(f \colon X \to K'\) inducing this isomorphism on \(\pi_n\text{.}\)

We extend \(f\) over the higher cells of \(K\) inductively. For each \((n+2)\)-cell of \(K\text{,}\) the composite of its attaching map with \(f\) lies in \(\pi_{n+1}(K') = 0\text{,}\) so it is null-homotopic and \(f\) extends over the cell. The same argument applies to every higher-dimensional cell, since \(\pi_i(K') = 0\) for all \(i > n\text{.}\) The resulting map \(f \colon K \to K'\) induces an isomorphism on \(\pi_n\) (by construction) and on every other homotopy group (both sides being zero), so \(f\) is a weak homotopy equivalence and hence, by Whitehead’s Theorem (Theorem 8.11), a homotopy equivalence.

Remark 13.14. Moore spaces.

There is a homology version of an Eilenberg–MacLane space, known as a Moore space. For an abelian group \(G\) and \(n \geq 1\) there exists a CW complex \(M(G, n)\) with

\begin{equation*} \widetilde{H}_k\bigl(M(G, n)\bigr) \cong \begin{cases} G & k = n, \\ 0 & \text{otherwise} \end{cases} \end{equation*}

(Exercise 13.19). Moore spaces can in fact be used to construct Eilenberg–MacLane spaces. For a based space \(X\text{,}\) let \(\mathrm{SP}^m(X) := X^{\times m}/\Sigma_m\) be the \(m\)th symmetric product, and let the infinite symmetric product \(\mathrm{SP}^\infty(X)\) be the colimit of the \(\mathrm{SP}^m(X)\) along the maps that add the basepoint as an extra coordinate. The Dold–Thom theorem says that for any connected CW complex \(X\) we have \(\pi_n\bigl(\mathrm{SP}^\infty(X)\bigr) \cong \widetilde{H}_n(X)\) for all \(n\text{.}\) In particular, \(\mathrm{SP}^\infty\bigl(M(G, n)\bigr)\) is a \(K(G, n)\text{.}\)

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