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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $X$ be an [[absolute neighborhood retract|DK3MT3HYX5]] of [[covering dimension|A3W7RGXXK9]] $n$.\nSuppose that $X$ is homogeneous: for all points $x, y \\in X$ there is a [[homeomorphism|0QT00PB5G2]] $h \\colon X \\to X$ with $h(x) = y$.\nThen $X$ is a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.\n\n## Remarks\n\nEvery connected topological manifold is a homogeneous absolute neighborhood retract, so the conjecture asks for a converse.\nThe conjecture is known to hold for $n = 1$ and $n = 2$.\nIn dimension $3$, it is known to imply the Poincaré conjecture.\nIn general it is open.\nThe finite dimension is needed: the Hilbert cube is a homogeneous compact absolute retract of infinite dimension that is not a manifold of any finite dimension.\n\n## Explanations\n\n### L1 human\n\nA manifold looks the same at every point, in the sense that any point can be moved to any other by a self-homeomorphism.\nIt is also a well-behaved space in the sense of homotopy theory.\nThe conjecture says that these two properties, in finite dimension, are enough to force a space to be a manifold.\n",
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  "front": {
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    "history": {
      "first_stated": "1965",
      "posed_by": "R. H. Bing and Karol Borsuk"
    },
    "kind": "conjecture",
    "msc": [
      "57P05",
      "55M15"
    ],
    "name": "Bing–Borsuk conjecture",
    "same_as": {
      "wikidata": "Q55605472",
      "wikipedia": "Bing–Borsuk conjecture"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bing%E2%80%93Borsuk_conjecture"
      }
    ],
    "tag": "VK39X07EYK",
    "uses": [
      "DK3MT3HYX5",
      "0QT00PB5G2",
      "A3W7RGXXK9",
      "QQQ9B1DK4P"
    ]
  },
  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/VK39X07EYK/",
  "links": [
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      "field": "about",
      "target": "DK3MT3HYX5"
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  "markdown": "https://math2.tamnd.com/node/VK39X07EYK.md",
  "msc": [
    "57P05",
    "55M15"
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  "name": "Bing–Borsuk conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VK3/VK39X07EYK.md",
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  "tag": "VK39X07EYK",
  "uri": "m2://VK39X07EYK",
  "url": "https://math2.tamnd.com/wiki/bing-borsuk-conjecture/",
  "writing": {
    "tag": "VK39X07EYK",
    "kind": "conjecture",
    "level": "W0",
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        "note": "the row of the inventory that the node accounts for",
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