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?
closed as off topic by Franz Lemmermeyer, J.C. Ottem, Andrew Stacey, Qiaochu Yuan, S. Carnahan♦ Apr 29 '11 at 10:06
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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).