{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(X, x_0)$ be a [[pointed space|T9DG8DDE75]], and let $p, q \\ge 1$.\nGive $S^p \\times S^q$ the [[CW structure|5W95QFHXPR]] with cells of dimensions $0$, $p$, $q$ and $p + q$, so that its $(p+q-1)$-skeleton is the [[wedge sum|5JSWM1HBX2]] $S^p \\vee S^q$.\nLet $w \\colon S^{p+q-1} \\to S^p \\vee S^q$ be the attaching map of the top cell.\nFor $\\alpha = [f] \\in \\pi_p(X, x_0)$ and $\\beta = [g] \\in \\pi_q(X, x_0)$, the *Whitehead product* is\n$$[\\alpha, \\beta] = [(f \\vee g) \\circ w] \\in \\pi_{p+q-1}(X, x_0),$$\nwhere $f \\vee g \\colon S^p \\vee S^q \\to X$ is $f$ on $S^p$ and $g$ on $S^q$, and $\\pi_k$ are the [[homotopy groups|P976WYEFMB]].\n\n## Remarks\n\nThe class $[\\alpha, \\beta]$ depends only on $\\alpha$ and $\\beta$.\nFor $p = q = 1$, the map $w$ traces the boundary of the square in the torus, and $[\\alpha, \\beta]$ is the commutator $\\alpha \\beta \\alpha^{-1} \\beta^{-1}$, up to the choice of conventions.\n\nExample: let $\\iota_2 \\in \\pi_2(S^2)$ be the class of the identity.\nThen $[\\iota_2, \\iota_2] \\in \\pi_3(S^2) \\cong \\Z$ has Hopf invariant $\\pm 2$, so it is twice a generator, up to sign.\n\nNon-example of a nonzero product: if $X$ is an H-space, for instance a topological group, then all Whitehead products in $X$ vanish.\nIn particular, every Whitehead product in $S^1$ or $S^3$ is zero.\n\nFor $p, q \\ge 2$ the product is bilinear and satisfies $[\\alpha, \\beta] = (-1)^{pq} [\\beta, \\alpha]$, together with a graded Jacobi identity.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "QKEE8AT2NR",
    "kind": "definition",
    "msc": [
      "55Q15"
    ],
    "name": "Whitehead product",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Whitehead_product"
      }
    ],
    "tag": "WP12H9FG91",
    "uses": [
      "P976WYEFMB",
      "5JSWM1HBX2",
      "5W95QFHXPR",
      "T9DG8DDE75"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/WP12H9FG91/",
  "links": [
    {
      "field": "defines",
      "target": "QKEE8AT2NR"
    },
    {
      "field": "uses",
      "target": "P976WYEFMB"
    },
    {
      "field": "uses",
      "target": "5JSWM1HBX2"
    },
    {
      "field": "uses",
      "target": "5W95QFHXPR"
    },
    {
      "field": "uses",
      "target": "T9DG8DDE75"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/WP12H9FG91.md",
  "msc": [
    "55Q15"
  ],
  "name": "Whitehead product",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WP1/WP12H9FG91.md",
  "status": null,
  "tag": "WP12H9FG91",
  "uri": "m2://WP12H9FG91",
  "url": "https://math2.tamnd.com/wiki/whitehead-product-definition/",
  "writing": {
    "tag": "WP12H9FG91",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}