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  "body": "\n## Definition\n\nLet $X$ be a [[path-connected space|0WB6DH499T]] with base point, and let $\\pi_k$ denote the $k$-th [[homotopy group|P976WYEFMB]].\nA *Postnikov tower* of $X$ is a sequence of pointed path-connected spaces $X_n$, $n \\ge 1$, with pointed maps $f_n \\colon X \\to X_n$ and $p_{n} \\colon X_{n} \\to X_{n-1}$ for $n \\ge 2$, such that:\n\n1. $p_n \\circ f_n = f_{n-1}$ for all $n \\ge 2$.\n2. $f_n$ induces an isomorphism $\\pi_k(X) \\to \\pi_k(X_n)$ for all $k \\le n$.\n3. $\\pi_k(X_n) = 0$ for all $k \u003e n$.\n\n## Remarks\n\nEvery path-connected CW complex has a Postnikov tower, built by attaching cells of dimension at least $n + 2$ to $X$ to kill $\\pi_k$ for $k \u003e n$.\nUp to homotopy, $p_n$ can be taken to be a fibration, and then its fibre is an [[Eilenberg–MacLane space|AJHEYJE5XY]] $K(\\pi_n(X), n)$.\nThe map from $X$ to the homotopy limit of the tower is a weak homotopy equivalence.\n\nExample: let $X = K(G, m)$.\nTake $X_n$ a point for $n \u003c m$ and $X_n = X$ for $n \\ge m$, with the evident maps.\nClause 2 holds since $\\pi_k(X) = 0$ for $k \u003c m$, and clause 3 holds since $\\pi_k(X) = 0$ for $k \u003e m$.\n\nNon-example: for $X = S^2$, the choice $X_2 = S^2$ with $f_2 = \\id$ fails clause 3, since $\\pi_3(S^2) \\cong \\Z \\ne 0$.\nA valid choice is $X_2 = \\C P^\\infty$, a $K(\\Z, 2)$.\n",
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