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  "body": "\n## Problem\n\nLet $n \\ge 0$ and let $M$ be an $n$-dimensional [[topological manifold|QQQ9B1DK4P]].\nDoes $M$ admit a [[triangulation|YH170WBQ6M]]?\nThat is, is there a [[simplicial complex|V8GZ0W1ZZ5]] $K$ whose geometric realization $\\abs{K}$ is [[homeomorphic|0QT00PB5G2]] to $M$?\nThe triangulation conjecture asserts that the answer is yes for every manifold.\n\n## Remarks\n\nThe question was raised by Hellmuth Kneser in 1926.\nThe conjecture holds in dimensions $n \\le 3$.\nFor $n \\le 1$ this is elementary, for $n = 2$ it is [[Radó's theorem|C9X05EDS5F]], and for $n = 3$ it follows from [[Moise's theorem|SV0816DJP9]].\nIt fails in dimension $4$: the closed [[$E_8$ manifold|BNDRN0G9BR]] of Michael Freedman is not triangulable, which follows from properties of the [[Casson invariant|33GY6P5A4T]].\nIt fails in every dimension $n \\ge 5$.\nGalewski and Stern, and independently Matumoto, reduced this case to a question about [[homology 3-spheres|HK8NS02NNE]] and homology cobordism.\nCiprian Manolescu answered that question in 2013 with $\\mathrm{Pin}(2)$-equivariant Seiberg–Witten Floer homology, which gives closed non-triangulable manifolds of each dimension $n \\ge 5$.\nThe conjecture asks for any simplicial triangulation, not one that is combinatorial.\nThe stronger question whether every manifold has a piecewise linear structure was settled negatively earlier, by Kirby and Siebenmann in 1969, through the [[Kirby–Siebenmann class|E4NJER1RFQ]].\nEvery smooth manifold is triangulable, so the counterexamples are manifolds without a smooth structure.\n\n## Explanations\n\n### L1 human\n\nA triangulation cuts a space into points, segments, triangles, tetrahedra and their higher analogues, glued along common faces.\nSurfaces and three-dimensional manifolds can always be cut up this way.\nThe conjecture said that every manifold, in every dimension, can be cut up like this.\nIt turned out to be false: in dimension four and in every higher dimension there are manifolds that admit no such decomposition at all.\n",
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      "first_stated": "1926",
      "posed_by": "Hellmuth Kneser"
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    "kind": "problem",
    "msc": [
      "57Q15",
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    "name": "Triangulation conjecture",
    "sources": [
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        "url": "https://en.wikipedia.org/wiki/Triangulation_(topology)"
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        "url": "https://arxiv.org/abs/1607.08163"
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  "name": "Triangulation conjecture",
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