{
  "aliases": null,
  "backlinks": null,
  "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 *Whitehead tower* of $X$ is a sequence of pointed spaces $X\\langle n \\rangle$, $n \\ge 0$, with $X\\langle 0 \\rangle = X$.\nIt comes with pointed maps $q_n \\colon X\\langle n \\rangle \\to X\\langle n - 1 \\rangle$ for $n \\ge 1$.\nFor each $n \\ge 0$, the composite $r_n = q_1 \\circ \\cdots \\circ q_n \\colon X\\langle n \\rangle \\to X$ must satisfy:\n\n1. $X\\langle n \\rangle$ is [[$n$-connected|1A943BWBQT]], that is, $\\pi_k(X\\langle n \\rangle) = 0$ for all $k \\le n$.\n2. $r_n$ induces an isomorphism $\\pi_k(X\\langle n \\rangle) \\to \\pi_k(X)$ for all $k \u003e n$.\n\nThe space $X\\langle n \\rangle$ is the *$n$-connected cover* of $X$.\n\n## Remarks\n\nFor a CW complex $X$ a Whitehead tower exists and each $X\\langle n \\rangle$ is unique up to weak homotopy equivalence over $X$.\nUp to homotopy, $q_n$ can be taken to be a fibration, and then its fibre is an [[Eilenberg–MacLane space|AJHEYJE5XY]] $K(\\pi_n(X), n - 1)$.\nFor $X$ a CW complex, $X\\langle 1 \\rangle \\to X$ is the [[universal cover|Y1YPE6PM52]].\nThe tower is dual to the [[Postnikov tower|EKGEZGHYBW]]: $X\\langle n \\rangle$ is the homotopy fibre of $X \\to X_n$.\n\nExample: for $X = S^3$, $X\\langle 1 \\rangle = X\\langle 2 \\rangle = S^3$, since $S^3$ is $2$-connected.\nThe cover $S^3\\langle 3 \\rangle \\to S^3$ is a fibration with fibre $K(\\Z, 2) = \\C P^\\infty$.\n\nNon-example: for $X = S^1$, the identity $S^1 \\to S^1$ as $X\\langle 1 \\rangle$ fails clause 1, since $\\pi_1(S^1) \\cong \\Z \\ne 0$.\nThe universal cover $\\R \\to S^1$ satisfies both clauses.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "D3649AJ30F",
    "kind": "definition",
    "msc": [
      "55S45",
      "55P20"
    ],
    "name": "Whitehead tower",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Postnikov_system"
      }
    ],
    "tag": "ZYCX4PN96G",
    "uses": [
      "0WB6DH499T",
      "P976WYEFMB",
      "1A943BWBQT",
      "AJHEYJE5XY",
      "Y1YPE6PM52",
      "EKGEZGHYBW"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/ZYCX4PN96G/",
  "links": [
    {
      "field": "defines",
      "target": "D3649AJ30F"
    },
    {
      "field": "uses",
      "target": "0WB6DH499T"
    },
    {
      "field": "uses",
      "target": "P976WYEFMB"
    },
    {
      "field": "uses",
      "target": "1A943BWBQT"
    },
    {
      "field": "uses",
      "target": "AJHEYJE5XY"
    },
    {
      "field": "uses",
      "target": "Y1YPE6PM52"
    },
    {
      "field": "uses",
      "target": "EKGEZGHYBW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/ZYCX4PN96G.md",
  "msc": [
    "55S45",
    "55P20"
  ],
  "name": "Whitehead tower",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZYC/ZYCX4PN96G.md",
  "status": null,
  "tag": "ZYCX4PN96G",
  "uri": "m2://ZYCX4PN96G",
  "url": "https://math2.tamnd.com/wiki/whitehead-tower-definition/",
  "writing": {
    "tag": "ZYCX4PN96G",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}