# Is every manifold triangulable?

by quasar987
Tags: manifold, triangulable
 Sci Advisor HW Helper PF Gold P: 4,768 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?
P: 1,716
 Quote by quasar987 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?
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.
 P: 336 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.
P: 1,716

## Is every manifold triangulable?

 Quote by quasar987 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?
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
 Sci Advisor HW Helper P: 9,421 it seems indeed to depend on your definition of triangulation. in gneral the answer may be unknown. consult: the history of topology by ioan mackenzie (or ask ron stern) http://books.google.com/books?id=7iR...ble%3F&f=false