In homotopy theory, an Eilenberg–MacLane space, denoted $K(G,n)$, is a connected topological space with homotopy groups are all trivial except for a single non-trivial homotopy group in dimension $n$.
$$\pi_n(K(G,n)) \cong G$$
For example a circle $S^1$ is $K(\mathbb{Z}, 1)$ since the the circle has a contractible universal cover and all higher homotopy groups vanish. It is not so easy to find Eilenberg-MacLane spaces within the class of finite cell complexes: existence results, though constructive, yield infinite-dimensional spaces as examples. Eilenberg-MacLane spaces of type $K(G, 1)$ include:
- All knot complements in $S^3$
- Configuration spaces $\mathcal{C}^n(\mathbb{R}^2)$ of points in the plane
- All state complexes; hence configuration spaces of graphs.
These spaces serve as the bridge to a surprising relationship between cohomology and homotopy: cohomology of any cellular space $X$ is expressible in terms of homotopy classes of maps of $X$ into Eilenberg-MacLane spaces, specifically:
$$H^n(X; G) \cong [X, K(G, n)]$$
Where $[X,Y]$ denotes a group of basepoint-preserving homotopy classes of maps $X \to Y$. One simple example is that $\pi_1(X) \cong [S^1, X]$ while $H^1(X;\mathbb{Z}) \cong [X, S^1]$.
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