Is every manifold triangulable?

1. Sep 22, 2010

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?

2. Sep 22, 2010

lavinia

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.

3. Sep 22, 2010

Eynstone

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.

4. Sep 22, 2010

lavinia

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

5. Sep 23, 2010

6. Oct 5, 2010

quasar987

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"