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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $G$ be a [[group|G5S1M9]], [[abelian|082GHNPJ53]] if $n \\ge 2$.\nAn *Eilenberg–MacLane space of type $K(G, n)$* is a [[path-connected space|0WB6DH499T]] $X$ with base point $x_0$ such that\n$$\\pi_n(X, x_0) \\cong G \\quad \\text{and} \\quad \\pi_k(X, x_0) = 0 \\text{ for all } k \\ne n, \\ k \\ge 1,$$\nwhere $\\pi_k$ is the $k$-th [[homotopy group|P976WYEFMB]].\n\n## Remarks\n\nThe condition that $G$ be abelian for $n \\ge 2$ is needed, since $\\pi_n$ is abelian for $n \\ge 2$.\nFor 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.\nOne often writes $K(G, n)$ for any such CW complex.\n\nExample: the circle $S^1$ is a $K(\\Z, 1)$.\nIt 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.\n\nNon-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 $K(\\Z, 2)$.\n",
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    "name": "Eilenberg–MacLane space",
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