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  "body": "\n## Statement\n\nLet $M$ be a closed [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]] that is [[homotopy equivalent|XZG6MTSWCN]] to the [[$4$-sphere|T62369RFH6]] $S^4$.\nThen $M$ is [[diffeomorphic|0KZZ70ZHXP]] to $S^4$ with its standard [[smooth structure|R2TQ82PCNB]].\n\n## Remarks\n\nThe conjecture is open.\nIts topological version is true: Michael Freedman proved in 1982 that such an $M$ is homeomorphic to $S^4$.\nSo the conjecture is equivalent to the statement that $S^4$ has no exotic smooth structure.\nIn dimensions $1$, $2$, $3$, $5$ and $6$ the smooth version holds, while in dimension $7$ it fails, by the exotic spheres of John Milnor.\nSome proposed counterexamples, among them an infinite family of Cappell–Shaneson homotopy spheres, have been shown to be standard.\nGauge-theoretic invariants such as the [[Seiberg–Witten invariants|PRDDMSTR4X]] need $b_2^+ \\ge 1$, so they say nothing about homotopy $4$-spheres.\n\n## Explanations\n\n### L1 human\n\nA homotopy sphere is a space that cannot be told apart from a sphere by the methods of algebraic topology.\nIn dimension four, every such space is known to be a sphere up to continuous deformation.\nThe open question is whether it is always a sphere in the smooth sense, where one also keeps track of calculus.\n",
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      "posed_by": "not attributed to one person; it is the smooth case in dimension 4 of the generalized Poincaré conjecture"
    },
    "kind": "conjecture",
    "msc": [
      "57R60",
      "57R55",
      "57K40"
    ],
    "name": "Smooth four-dimensional Poincaré conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture"
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    "tag": "P70VGYNFS5",
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      "F45VQ47GPV",
      "T62369RFH6",
      "XZG6MTSWCN",
      "0KZZ70ZHXP",
      "R2TQ82PCNB"
    ]
  },
  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/P70VGYNFS5/",
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  "msc": [
    "57R60",
    "57R55",
    "57K40"
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  "name": "Smooth four-dimensional Poincaré conjecture",
  "prov": null,
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/P70/P70VGYNFS5.md",
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  "tag": "P70VGYNFS5",
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  "url": "https://math2.tamnd.com/wiki/smooth-four-dimensional-poincare-conjecture/",
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    "tag": "P70VGYNFS5",
    "kind": "conjecture",
    "level": "W0",
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