{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "ZJVKZ8D64P"
    }
  ],
  "body": "\n## Example\n\nLet $S^3 = \\set{(z_0, z_1) \\in \\C^2 : \\abs{z_0}^2 + \\abs{z_1}^2 = 1}$, and identify $\\R^3$ with $\\C \\times \\R$.\nDefine $h \\colon S^3 \\to \\R^3$ by\n$$h(z_0, z_1) = \\bigl(2 z_0 \\overline{z_1}, \\ \\abs{z_0}^2 - \\abs{z_1}^2\\bigr).$$\nThe image lies in the unit sphere $S^2$, because\n$$\\abs{2 z_0 \\overline{z_1}}^2 + \\bigl(\\abs{z_0}^2 - \\abs{z_1}^2\\bigr)^2 = \\bigl(\\abs{z_0}^2 + \\abs{z_1}^2\\bigr)^2 = 1.$$\nTwo points of $S^3$ have the same image if and only if one is $(\\lambda z_0, \\lambda z_1)$ for the other $(z_0, z_1)$ and some $\\lambda \\in \\C$ with $\\abs{\\lambda} = 1$.\nSo each fibre of $h \\colon S^3 \\to S^2$ is a circle.\nThe map $h$ is a fibre bundle with fibre $S^1$, hence a [[Serre fibration|PB9JXQ2KC3]].\nEquivalently, $h$ is the quotient map $S^3 \\to \\C P^1 \\cong S^2$, $(z_0, z_1) \\mapsto [z_0 : z_1]$.\n\n## Remarks\n\nThe [[long exact sequence of homotopy groups|E0RV5D1GQ0]] of $S^1 \\to S^3 \\to S^2$ gives $\\pi_2(S^2) \\cong \\Z$ and $\\pi_3(S^2) \\cong \\pi_3(S^3) \\cong \\Z$.\nThe class of $h$ generates $\\pi_3(S^2)$, and $h$ has [[Hopf invariant|TEGTSY7XMX]] $\\pm 1$.\nAny two fibres are linked circles in $S^3$, with linking number $\\pm 1$.\nSimilar fibrations $S^3 \\to S^7 \\to S^4$ and $S^7 \\to S^{15} \\to S^8$ come from the quaternions and the octonions.\n\n## Explanations\n\n### L1 human\n\nThe Hopf fibration splits the $3$-sphere into circles, one circle over each point of the ordinary sphere.\nNo two of these circles are separate: every pair is linked like two rings of a chain.\nIt was the first example of a map from a sphere to a lower-dimensional sphere that cannot be deformed to a constant map.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "instance_of": [
      "PB9JXQ2KC3"
    ],
    "kind": "example",
    "msc": [
      "55R10",
      "55Q40",
      "55R05"
    ],
    "name": "Hopf fibration",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hopf_fibration"
      }
    ],
    "tag": "M5AKNYC4QQ"
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/M5AKNYC4QQ/",
  "links": [
    {
      "field": "instance_of",
      "target": "PB9JXQ2KC3"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/M5AKNYC4QQ.md",
  "msc": [
    "55R10",
    "55Q40",
    "55R05"
  ],
  "name": "Hopf fibration",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M5A/M5AKNYC4QQ.md",
  "status": null,
  "tag": "M5AKNYC4QQ",
  "uri": "m2://M5AKNYC4QQ",
  "url": "https://math2.tamnd.com/wiki/hopf-fibration/",
  "writing": {
    "tag": "M5AKNYC4QQ",
    "kind": "example",
    "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
      }
    ]
  }
}