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Is every manifold triangulable?

  1. quasar987

    quasar987 4,770
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    In Lee's Intro to topological manifolds, p.105, it is written that every manifold of dimension 3 or below is triangulable. But in dimension 4, threre are known examples of non triangulable manifolds. In dimensions greater than four, the answer is unknown.

    But in Bott-Tu p.190, it is written that every manifold admits a triangulation.

    Which is right?
     
  2. jcsd
  3. lavinia

    lavinia 2,036
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    I would guess that Bott and Tu mean every smooth manifold since their book is about differential topology. It is a theorem of Whitehead, I believe, that every smooth manifold has a smooth triangulation.
     
  4. It seems that the authors probably have different definitions of triangulation.
    In my opinion, the problem boils down to whether every interval isomorphic to some interval in R^n is triangulable & hence the second statement looks good.
     
  5. lavinia

    lavinia 2,036
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    R. ]Kirby and L. C. Siebenmann, On the triangulation of manifolds and the hauptvermutung,
    Bull. Amer. Math. Soc., 75 (1969), 742-749.

    This paper is said to have an example of a non-triangulable 6 manifold
     
  6. mathwonk

    mathwonk 9,691
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  7. quasar987

    quasar987 4,770
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    In Whitney's geometric integration p.124, he credits S.S. Cains (1934) with Whitehead (1940) giving an improvement of the proof in "On C¹ complexes, Annals of Math. 41"
     
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