{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *$4$-manifold* is a [[topological manifold|QQQ9B1DK4P]] of dimension $4$, that is, a topological space $M$ such that:\n\n1. $M$ is Hausdorff,\n2. $M$ is second countable, and\n3. every point of $M$ has an open neighborhood homeomorphic to an open subset of $\\R^4$.\n\nA *smooth $4$-manifold* is a $4$-manifold with a [[smooth structure|R2TQ82PCNB]].\n\n## Remarks\n\nExample: the $4$-sphere $S^4 = \\set{x \\in \\R^5 : \\abs{x} = 1}$ is Hausdorff and second countable as a subspace of $\\R^5$, and stereographic projection from a point $p$ maps $S^4 \\setminus \\set{p}$ homeomorphically onto $\\R^4$, which gives clause 3.\n\nNon-example: take two copies of $\\R^4$ and glue them along the complement of the origin.\nEvery point has a neighborhood homeomorphic to $\\R^4$, but the two origins have no disjoint neighborhoods, so clause 1 fails.\n\nBy invariance of domain, no nonempty open subset of $\\R^4$ is homeomorphic to an open subset of $\\R^n$ with $n \\ne 4$, so the dimension is well defined.\nThe [[$E_8$ manifold|BNDRN0G9BR]] is a closed $4$-manifold with no smooth structure.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "8H58Q6YJCH",
    "kind": "definition",
    "msc": [
      "57K40"
    ],
    "name": "4-manifold",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/4-manifold"
      }
    ],
    "tag": "T4ESER8G7G",
    "uses": [
      "QQQ9B1DK4P",
      "R2TQ82PCNB"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/T4ESER8G7G/",
  "links": [
    {
      "field": "defines",
      "target": "8H58Q6YJCH"
    },
    {
      "field": "uses",
      "target": "QQQ9B1DK4P"
    },
    {
      "field": "uses",
      "target": "R2TQ82PCNB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/T4ESER8G7G.md",
  "msc": [
    "57K40"
  ],
  "name": "4-manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/T4E/T4ESER8G7G.md",
  "status": null,
  "tag": "T4ESER8G7G",
  "uri": "m2://T4ESER8G7G",
  "url": "https://math2.tamnd.com/wiki/4-manifold-definition/",
  "writing": {
    "tag": "T4ESER8G7G",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}