{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $M$ be a closed connected [[3-manifold|R20TB2KEQQ]] whose [[fundamental group|S1XK3HXHFX]] is trivial.\nThen $M$ is [[homeomorphic|0QT00PB5G2]] to the [[3-sphere|T62369RFH6]] $S^3$.\n\n## Remarks\n\nPoincaré asked this question in 1904.\nPerelman proved it in 2002 and 2003, following Hamilton's program of Ricci flow.\nOne runs the Ricci flow from any Riemannian metric on $M$ and cuts out regions where the curvature blows up (surgery).\nFor simply connected $M$ the flow with surgery becomes extinct in finite time, and this shows that $M$ is a sphere.\nIt is also a special case of the [[geometrization conjecture|B27HB43K86]].\nBoth hypotheses matter: $\\R^3$ is simply connected but not closed, and the [[Poincaré homology sphere|NHERCK7C43]] is closed with the homology of $S^3$ but has nontrivial fundamental group.\nThe analogous statement in dimensions $n \\ge 5$ is due to Smale and in dimension $4$ to Freedman, in their topological versions.\n\n## Explanations\n\n### L1 human\n\nA closed $3$-dimensional space in which every loop can be shrunk to a point must be the $3$-sphere.\nIt was the most famous open problem of topology for a century.\nIt was one of the Clay Millennium Prize problems.\n\n### L2 human\n\nThe statement is a recognition theorem: a single algebraic condition identifies $S^3$ among all closed $3$-manifolds.\nIn practice it is now used as part of geometrization, which gives every closed $3$-manifold with finite fundamental group the form $S^3 / \\Gamma$ for a finite group $\\Gamma$ of isometries.\nIn the smooth category, it also follows that every smooth homotopy $3$-sphere is diffeomorphic to $S^3$, because topological $3$-manifolds have unique smooth structures.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "R20TB2KEQQ",
      "T62369RFH6"
    ],
    "kind": "statement",
    "msc": [
      "57K30",
      "57R60",
      "53E20"
    ],
    "name": "Poincaré conjecture",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q203586",
      "wikipedia": "Poincaré conjecture"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture"
      }
    ],
    "tag": "3R064KFF68",
    "uses": [
      "R20TB2KEQQ",
      "T62369RFH6",
      "S1XK3HXHFX",
      "0QT00PB5G2"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/3R064KFF68/",
  "links": [
    {
      "field": "about",
      "target": "R20TB2KEQQ"
    },
    {
      "field": "about",
      "target": "T62369RFH6"
    },
    {
      "field": "uses",
      "target": "R20TB2KEQQ"
    },
    {
      "field": "uses",
      "target": "T62369RFH6"
    },
    {
      "field": "uses",
      "target": "S1XK3HXHFX"
    },
    {
      "field": "uses",
      "target": "0QT00PB5G2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/3R064KFF68.md",
  "msc": [
    "57K30",
    "57R60",
    "53E20"
  ],
  "name": "Poincaré conjecture",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3R0/3R064KFF68.md",
  "status": null,
  "tag": "3R064KFF68",
  "uri": "m2://3R064KFF68",
  "url": "https://math2.tamnd.com/wiki/poincare-conjecture/",
  "writing": {
    "tag": "3R064KFF68",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}