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One of my classmates was telling me that it is an open question whether every 3-manifold can be triangulated. This was rather surprising. He said that the question as far as he remember is settled only for 4-manifold where answer is negative. If this is the case, can somebody shed some light why this problem is so hard?

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    $\begingroup$ This is not the place for questions that google can answer in 2 seconds. $\endgroup$
    – Franz Lemmermeyer
    Commented Apr 29, 2011 at 8:43
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    $\begingroup$ Sadly, it DOES appear to be the place. $\endgroup$
    – Igor Rivin
    Commented Apr 29, 2011 at 13:28

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Every $3$-manifold is triangulable.

This was proven by Edwin E. Moise in is paper "Affine structure in $3$-manifolds", Annals of Math. 56 (1952).

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    – S. Carnahan ♦
    Commented Apr 29, 2011 at 10:07

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