Definition 13.1.
A path-connected space \(X\) whose only nontrivial homotopy group is \(\pi_n(X) = G\) is called an Eilenberg–MacLane space, denoted \(K(G, n)\text{.}\)
Introduction
The motivating question for this chapter is the following. Given an abelian group \(G\) and a positive integer \(n\text{,}\) can we construct a space whose only nontrivial homotopy group is \(\pi_n = G\text{?}\) The answer is yes, and these spaces, called Eilenberg–MacLane spaces, will be central to the rest of the course.
Example 13.2.
We have already met several examples.
\begin{equation*}
S^1 = K(\mathbb{Z}, 1), \qquad \mathbb{C}P^\infty = K(\mathbb{Z}, 2), \qquad \mathbb{R}P^\infty = K(\mathbb{Z}/2, 1).
\end{equation*}
The first holds because \(\pi_1(S^1) \cong \mathbb{Z}\) (Theorem 2.16) and \(\pi_n(S^1) = 0\) for \(n \geq 2\text{,}\) using the covering \(\mathbb{R} \to S^1\) (Proposition 5.14). The latter two follow from the long exact sequences of the bundles \(S^1 \to S^\infty \to \mathbb{C}P^\infty\) (Example 10.15) and \(\mathbb{Z}/2 \to S^\infty \to \mathbb{R}P^\infty\) (Exercise 10.32) together with the contractibility of \(S^\infty\text{.}\)
Remark 13.3.
We have not yet shown that Eilenberg–MacLane spaces exist in any generality. Since \(\pi_n\) is abelian for \(n \geq 2\text{,}\) for \(n \geq 2\) a \(K(G, n)\) can only exist if \(G\) is abelian. The goal of this chapter is to show that this is the only obstruction. For any group \(G\text{,}\) abelian if \(n \geq 2\text{,}\) the Eilenberg–MacLane space \(K(G, n)\) exists, and it is unique up to homotopy equivalence among CW complexes. In general we must build it as a CW complex by hand.
