{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "7KEDZ9TFEE"
    },
    {
      "field": "uses",
      "target": "9VY90HJVZX"
    },
    {
      "field": "uses",
      "target": "DX3DJY91QH"
    },
    {
      "field": "uses",
      "target": "NFTS5A3NXE"
    },
    {
      "field": "uses",
      "target": "NWXJACYJC5"
    },
    {
      "field": "about",
      "target": "9VY90HJVZX"
    },
    {
      "field": "defines",
      "target": "K99YMX66XD"
    }
  ],
  "body": "\n## Summary\n\nThe fundamental class $[M]$ of a closed oriented $n$-manifold $M$ is a class in $H_n(M; \\Z)$ that restricts to the chosen local [[orientation|913WEQERBE]] at every point.\nIt represents the manifold itself as a single homology cycle.\nCapping with it gives Poincaré duality, and pairing top-degree cohomology classes with it gives numbers such as degrees of maps and characteristic numbers.\n\n## Explanations\n\n### L1 human\n\nCut a closed oriented surface into triangles and orient them all the same way.\nThe sum of all these triangles is a cycle, because each edge appears twice with opposite signs and cancels.\nThis cycle is the fundamental class: the whole surface counted once.\nFor a connected closed oriented manifold it generates the top homology group, which is $\\Z$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "57N65",
      "55N10"
    ],
    "name": "Fundamental class",
    "same_as": {
      "wikidata": "Q4493388",
      "wikipedia": "Fundamental class"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fundamental_class"
      }
    ],
    "tag": "ZEA0PEM3B2"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/ZEA0PEM3B2/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/ZEA0PEM3B2.md",
  "msc": [
    "57N65",
    "55N10"
  ],
  "name": "Fundamental class",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZEA/ZEA0PEM3B2.md",
  "status": null,
  "tag": "ZEA0PEM3B2",
  "uri": "m2://ZEA0PEM3B2",
  "url": "https://math2.tamnd.com/wiki/fundamental-class/",
  "writing": {
    "tag": "ZEA0PEM3B2",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}