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  "body": "\n## Problem\n\nLet $K$ and $L$ be [[simplicial complexes|V8GZ0W1ZZ5]] whose geometric realizations $\\abs{K}$ and $\\abs{L}$ are [[homeomorphic|0QT00PB5G2]].\nDo $K$ and $L$ always have subdivisions $K'$ and $L'$ that are isomorphic as simplicial complexes?\n\nEquivalently: are any two [[triangulations|YH170WBQ6M]] of the same space related by a piecewise linear homeomorphism?\n\n## Remarks\n\nThe name is German for \"main conjecture\".\nA positive answer would make the combinatorial invariants of a triangulation, such as simplicial homology, automatically topological invariants; homology was later shown to be a topological invariant by other means.\nThe answer is yes for complexes of dimension at most $2$, and for $3$-manifolds by [[Moise's theorem|SV0816DJP9]]; for surfaces it goes with [[Radó's theorem|C9X05EDS5F]].\nThe answer is no in general: John Milnor gave counterexamples in 1961 among finite simplicial complexes that are not manifolds.\nFor manifolds the answer is also no.\nIn dimensions at least $5$, the theory behind the [[Kirby–Siebenmann class|E4NJER1RFQ]] gives topological manifolds with several PL structures.\nThe corresponding triangulations are not related by any piecewise linear homeomorphism.\n\n## Explanations\n\n### L1 human\n\nThe same shape can be cut into triangles, or into higher-dimensional simplices, in many ways.\nThe question is whether any two such cuttings can be refined until they become the same cutting.\nFor shapes of dimension up to three this works, but in higher dimensions there are shapes where it fails.\n",
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