Using the previous results, any abelian group \(G\) can be realized as \(\pi_n(X)\text{,}\) for \(n \geq 2\text{,}\) of some space \(X\text{.}\) Indeed, choose a presentation \(G = \langle g_\alpha \mid r_\beta \rangle\) as an abelian group, and take
\begin{equation*}
X = \Bigl(\bigvee_\alpha S^n_\alpha\Bigr) \cup \bigcup_\beta D^{n+1}_\beta,
\end{equation*}
with the spheres \(S^n_\alpha\) corresponding to the generators, and the discs attached by maps \(\varphi_\beta \colon S^n_\beta \to \bigvee_\alpha S^n_\alpha\) with \([\varphi_\beta] = r_\beta\) under Lemma 13.4. Then Lemma 13.7 says that \(\pi_n(X) \cong G\) and \(\pi_i(X) = 0\) for \(i < n\text{.}\) However, we have no control over the higher homotopy groups. To construct Eilenberg–MacLane spaces we have to kill them.
For \(n = 1\) the same construction works for any group \(G = \langle g_\alpha \mid r_\beta \rangle\text{,}\) not necessarily abelian, with each \(\varphi_\beta \colon S^1 \to \bigvee_\alpha S^1_\alpha\) the loop spelling out the word \(r_\beta\text{.}\) By the Seifert–Van Kampen theorem (Theorem 4.19), the resulting \(2\)-dimensional complex has \(\pi_1 \cong G\text{.}\)
