{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 0$ and let $M$ and $N$ be closed (compact, without boundary) [[smooth manifolds|F45VQ47GPV]] of dimension $n$.\nA *cobordism* between $M$ and $N$ is a compact smooth [[manifold with boundary|7DS6B2BFK8]] $W$ of dimension $n + 1$ with $\\partial W$ diffeomorphic to $M \\sqcup N$.\nThe manifolds $M$ and $N$ are *cobordant* if a cobordism between them exists.\n\n## Remarks\n\nBeing cobordant is an [[equivalence relation|0G8VEZ3WTK]]: $M \\times [0, 1]$ gives reflexivity, and two cobordisms can be glued along a common boundary component for transitivity.\nThe set of classes $\\Omega_n^O$ is an abelian group under disjoint union, and every element has order $2$, since $M \\sqcup M$ bounds $M \\times [0, 1]$.\nIn the oriented version, $W$ is oriented and $\\partial W \\cong M \\sqcup (-N)$, where $-N$ is $N$ with the opposite orientation; this gives the groups $\\Omega_n^{SO}$.\n\nExample: the circle $S^1$ bounds the disc $D^2$, so $S^1$ is cobordant to the empty manifold.\nEvery sphere $S^n = \\partial D^{n+1}$ is likewise null-cobordant.\n\nNon-example: the real projective plane $\\R P^2$ is not the boundary of any compact $3$-manifold.\nA closed manifold $M$ that bounds a compact manifold $W$ has even Euler characteristic, but $\\chi(\\R P^2) = 1$.\n\nA manifold cobordant to the empty manifold is called null-cobordant.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "T8BVGPPSZP",
    "kind": "definition",
    "msc": [
      "57R75",
      "55N22"
    ],
    "name": "Cobordism",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cobordism"
      }
    ],
    "tag": "WTRPSJKAZH",
    "uses": [
      "F45VQ47GPV",
      "7DS6B2BFK8",
      "0G8VEZ3WTK"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/WTRPSJKAZH/",
  "links": [
    {
      "field": "defines",
      "target": "T8BVGPPSZP"
    },
    {
      "field": "uses",
      "target": "F45VQ47GPV"
    },
    {
      "field": "uses",
      "target": "7DS6B2BFK8"
    },
    {
      "field": "uses",
      "target": "0G8VEZ3WTK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/WTRPSJKAZH.md",
  "msc": [
    "57R75",
    "55N22"
  ],
  "name": "Cobordism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WTR/WTRPSJKAZH.md",
  "status": null,
  "tag": "WTRPSJKAZH",
  "uri": "m2://WTRPSJKAZH",
  "url": "https://math2.tamnd.com/wiki/cobordism-definition/",
  "writing": {
    "tag": "WTRPSJKAZH",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}