Eilenberg–MacLane space

Builds on Group Abelian group Path-connected space Homotopy group

Definition

Let $n \ge 1$ and let $G$ be a group, abelian if $n \ge 2$ . An Eilenberg–MacLane space of type $K(G, n)$ is a path-connected space $X$ with base point $x_0$ such that $\pi_n(X, x_0) \cong G \quad \text{and} \quad \pi_k(X, x_0) = 0 \text{ for all } k \ne n, \ k \ge 1,$ where $\pi_k$ is the $k$ -th homotopy group.

Remarks

The condition that $G$ be abelian for $n \ge 2$ is needed, since $\pi_n$ is abelian for $n \ge 2$ . For every such $G$ and $n$ there is a CW complex of type $K(G, n)$ , and any two CW complexes of type $K(G, n)$ are homotopy equivalent. One often writes $K(G, n)$ for any such CW complex.

Example: the circle $S^1$ is a $K(\Z, 1)$ . It is path-connected, $\pi_1(S^1) \cong \Z$ , and $\pi_k(S^1) = 0$ for $k \ge 2$ because its universal cover $\R$ is contractible.

Non-example: the sphere $S^2$ is path-connected with $\pi_2(S^2) \cong \Z$ and $\pi_1(S^2) = 0$ , but $\pi_3(S^2) \cong \Z$ , so it is not a $K(\Z, 2)$ .

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Tag	`WAS9NHKMYD`
URI	`m2://WAS9NHKMYD`
Kind	definition
Writing	W0 stub
MSC	55P20
Defines	Eilenberg–MacLane space
Uses	Group, Abelian group, Path-connected space, Homotopy group
Changed	2 November 2026

Eilenberg–MacLane space

Definition

Remarks

Reading path

Further reading

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