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  "body": "\n## Statement\n\nLet $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $3$.\n\n1. $M$ has a [[triangulation|YH170WBQ6M]].\n2. Any two triangulations $h_1 \\colon \\abs{K_1} \\to M$ and $h_2 \\colon \\abs{K_2} \\to M$ are related by a piecewise linear homeomorphism. That is, $K_1$ and $K_2$ have subdivisions that are isomorphic as simplicial complexes.\n\n## Remarks\n\nEquivalently, every topological $3$-manifold has a [[piecewise linear structure|15MKGRNM9K]], unique up to piecewise linear homeomorphism.\nCombined with results comparing piecewise linear and smooth structures in low dimensions, it follows that every topological $3$-manifold has a smooth structure, unique up to diffeomorphism.\nEdwin Moise proved the theorem in 1952; R. H. Bing later gave another proof.\nA key step approximates a homeomorphism of $3$-manifolds by piecewise linear ones, which needs tameness results for surfaces in $3$-manifolds.\nThe statement fails in dimension $4$, where the $E_8$ manifold has no triangulation, and part 2 fails for manifolds in higher dimensions.\n\n## Explanations\n\n### L1 human\n\nEvery $3$-dimensional manifold can be cut into tetrahedra glued face to face, and in essentially only one way.\nSo in dimension three, the topological, the piecewise linear and the smooth points of view all describe the same manifolds.\n\n### L2 human\n\nThe theorem lets $3$-manifold topology use triangulations, normal surfaces and smooth tools freely, without losing generality.\nIt is the dimension-three case of the Hauptvermutung and of the triangulation conjecture, both of which hold in dimensions up to three and fail in general.\n",
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