{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "12BXVH5W5H"
    },
    {
      "field": "uses",
      "target": "34B7K77P31"
    },
    {
      "field": "uses",
      "target": "5DTY2FCT9Z"
    },
    {
      "field": "uses",
      "target": "90RA6TAT7H"
    },
    {
      "field": "uses",
      "target": "HSD6FKCYRC"
    },
    {
      "field": "uses",
      "target": "TEG1ZX1AGJ"
    },
    {
      "field": "about",
      "target": "34B7K77P31"
    },
    {
      "field": "about",
      "target": "90RA6TAT7H"
    },
    {
      "field": "about",
      "target": "HSD6FKCYRC"
    },
    {
      "field": "about",
      "target": "TEG1ZX1AGJ"
    },
    {
      "field": "defines",
      "target": "NFTS5A3NXE"
    }
  ],
  "body": "\n## Summary\n\nThe intersection form of a closed oriented $4$-manifold $M$ is a symmetric bilinear form on $H^2(M; \\Z)$.\nIt is the cup product followed by evaluation on the [[fundamental class|ZEA0PEM3B2]].\nGeometrically, it counts with signs the intersection points of two surfaces in $M$ in general position.\nBy Poincaré duality it is unimodular on $H^2(M; \\Z)$ modulo torsion.\nIt is the central invariant of $4$-manifold topology: Freedman's classification and Donaldson's theorem are statements about it.\n\n## Explanations\n\n### L1 human\n\nIn a $4$-dimensional space, two surfaces in general position meet in isolated points, just as two curves do in the plane.\nEach point counts as $+1$ or $-1$ according to orientations, and the total is the intersection number.\nThe intersection form records this number for every pair of surface classes.\nFor the complex projective plane, a line meets another line in exactly one point, so the form is the $1 \\times 1$ matrix $(1)$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "57K40",
      "57N65",
      "55N45"
    ],
    "name": "Intersection form",
    "same_as": {
      "wikidata": "Q6056717",
      "wikipedia": "Intersection form of a 4-manifold"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Intersection_form_of_a_4-manifold"
      }
    ],
    "tag": "WK8GKRKAVD"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/WK8GKRKAVD/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/WK8GKRKAVD.md",
  "msc": [
    "57K40",
    "57N65",
    "55N45"
  ],
  "name": "Intersection form",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WK8/WK8GKRKAVD.md",
  "status": null,
  "tag": "WK8GKRKAVD",
  "uri": "m2://WK8GKRKAVD",
  "url": "https://math2.tamnd.com/wiki/intersection-form/",
  "writing": {
    "tag": "WK8GKRKAVD",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}