{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 0$.\nLet $M$ and $N$ be closed [[smooth manifolds|F45VQ47GPV]] of dimension $n$, that is, compact and without boundary.\nThey are *cobordant* if there is a compact smooth [[manifold with boundary|7DS6B2BFK8]] $W$ of dimension $n + 1$ whose boundary is diffeomorphic to $M \\sqcup N$.\nThis is an equivalence relation; let $\\mathfrak{N}_n$ be the set of its classes.\nThe *unoriented cobordism ring* is the [[graded ring|CAPQBQDRA7]]\n$$\\mathfrak{N}_* = \\bigoplus_{n \\ge 0} \\mathfrak{N}_n,$$\nwith addition $[M] + [N] = [M \\sqcup N]$, zero the class of the empty manifold, and multiplication $[M] \\cdot [N] = [M \\times N]$.\n\n## Remarks\n\nEvery element has order $2$, since $M \\sqcup M$ is the boundary of $M \\times [0, 1]$; so $\\mathfrak{N}_*$ is an algebra over $\\F_2$.\nThom proved that $\\mathfrak{N}_* \\cong \\F_2[x_i : i \\ge 2,\\ i \\neq 2^j - 1]$ with $\\deg x_i = i$, and that $\\mathfrak{N}_n$ is isomorphic to the homotopy group $\\pi_n(MO)$ of the Thom spectrum $MO$.\n\nThe *oriented cobordism ring* $\\Omega^{SO}_*$ uses closed oriented manifolds and requires $\\partial W \\cong M \\sqcup (-N)$ as oriented manifolds, where $-N$ is $N$ with the opposite orientation.\nIt is graded commutative, and $\\Omega^{SO}_* \\otimes \\Q$ is a polynomial ring over $\\Q$ on the classes of the complex projective spaces $\\C P^{2k}$, $k \\ge 1$.\n\nExample: $\\mathfrak{N}_1 = 0$, since every closed $1$-manifold is a finite union of circles and each circle bounds a disc.\nAlso $\\mathfrak{N}_2 \\cong \\Z/2$, generated by $\\R P^2$, which does not bound because its Euler characteristic is odd.\n\nNon-example: $\\R P^2$ is a closed surface that is not cobordant to the empty manifold.\nIndeed, if $\\R P^2 = \\partial W$ for a compact $3$-manifold $W$, then the Euler characteristic of $\\R P^2$ would be even, but it is $1$.\n\nFurther variants use manifolds with a stably complex structure, a spin structure or a framing, and each is the coefficient ring of a Thom spectrum.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "C02WQY819R",
    "kind": "definition",
    "msc": [
      "57R75",
      "55N22"
    ],
    "name": "Cobordism ring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cobordism_ring"
      }
    ],
    "tag": "WEGT4EZCX2",
    "uses": [
      "T8BVGPPSZP",
      "F45VQ47GPV",
      "7DS6B2BFK8",
      "CAPQBQDRA7"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/WEGT4EZCX2/",
  "links": [
    {
      "field": "defines",
      "target": "C02WQY819R"
    },
    {
      "field": "uses",
      "target": "T8BVGPPSZP"
    },
    {
      "field": "uses",
      "target": "F45VQ47GPV"
    },
    {
      "field": "uses",
      "target": "7DS6B2BFK8"
    },
    {
      "field": "uses",
      "target": "CAPQBQDRA7"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/WEGT4EZCX2.md",
  "msc": [
    "57R75",
    "55N22"
  ],
  "name": "Cobordism ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WEG/WEGT4EZCX2.md",
  "status": null,
  "tag": "WEGT4EZCX2",
  "uri": "m2://WEGT4EZCX2",
  "url": "https://math2.tamnd.com/wiki/cobordism-ring-definition/",
  "writing": {
    "tag": "WEGT4EZCX2",
    "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
      }
    ]
  }
}