One of my classmates was telling me that it is an open question whether every 3manifold can be triangulated. This was rather surprising. He said that the question as far as he remember is settled only for 4manifold 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, Loop Space, Qiaochu Yuan, S. Carnahan♦ Apr 29 '11 at 10:06Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question. 


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). 

