{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\pi \\colon E \\to B$ be a real vector bundle of rank $n$ with a continuous [[inner product|0NB0AWGJBW]] on its fibers.\nLet $D(E) = \\set{v \\in E : \\abs{v} \\le 1}$ be the disc bundle and $S(E) = \\set{v \\in E : \\abs{v} = 1}$ the unit [[sphere bundle|3VQ2V5HH3R]].\nThe *Thom space* of $E$ is the [[quotient space|0BR3FY5Q8M]]\n$$\\operatorname{Th}(E) = D(E) / S(E),$$\nin which $S(E)$ is collapsed to one point, the base point.\n\n## Remarks\n\nUp to homeomorphism, $\\operatorname{Th}(E)$ does not depend on the inner product.\nIf $B$ is compact Hausdorff, $\\operatorname{Th}(E)$ is homeomorphic to the [[one-point compactification|0GWN3CMRK8]] of $E$.\nThe notations $T(E)$, $B^E$ and $M(E)$ are also used.\n\nExample: if $B$ is a point, then $E = \\R^n$, $D(E) = D^n$ and $S(E) = S^{n-1}$, so $\\operatorname{Th}(E) = D^n / S^{n-1} \\cong S^n$.\nMore generally, for the trivial bundle $B \\times \\R^n$ over a compact space $B$, the Thom space is $(B \\times D^n)/(B \\times S^{n-1})$, the $n$-fold reduced suspension of $B$ with a disjoint base point added.\nThe Thom space of the tautological line bundle over $\\R P^m$ is $\\R P^{m+1}$.\n\nNon-example: the disc bundle $D(E)$ without the collapse is not the Thom space; it deformation retracts onto the zero section, so it is homotopy equivalent to $B$.\nFor $B$ a point and $n \\ge 1$ this gives a disc, which is contractible, while $\\operatorname{Th}(E) = S^n$ is not.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "5MNYT3AFPH",
    "kind": "definition",
    "msc": [
      "55R25",
      "57R20"
    ],
    "name": "Thom space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Thom_space"
      }
    ],
    "tag": "5152FTFE13",
    "uses": [
      "3VQ2V5HH3R",
      "0BR3FY5Q8M",
      "0NB0AWGJBW",
      "0GWN3CMRK8"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/5152FTFE13/",
  "links": [
    {
      "field": "defines",
      "target": "5MNYT3AFPH"
    },
    {
      "field": "uses",
      "target": "3VQ2V5HH3R"
    },
    {
      "field": "uses",
      "target": "0BR3FY5Q8M"
    },
    {
      "field": "uses",
      "target": "0NB0AWGJBW"
    },
    {
      "field": "uses",
      "target": "0GWN3CMRK8"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/5152FTFE13.md",
  "msc": [
    "55R25",
    "57R20"
  ],
  "name": "Thom space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/515/5152FTFE13.md",
  "status": null,
  "tag": "5152FTFE13",
  "uri": "m2://5152FTFE13",
  "url": "https://math2.tamnd.com/wiki/thom-space-definition/",
  "writing": {
    "tag": "5152FTFE13",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}