I tried to summarize the relations between the following notions of: a manifold (smooth, topological and PL), simpilicial complex, CW complex. However I found some inconsistencies, which may be not a problem for a specialist who knows how to interpret certain assertions, but I'm not a specialist so I'm trying to pose statements precise. So I would like to pose some statements, asking whether they are true (surely some of them follow directly from the definition but there are many conventions and I'm a bit confused about them):
1. The manifold homeomorphic to simplicial complex is usual referred as `admiting triangulation'
2. a) Every PL manifold admits triangulation
b) every triangulated manifold is CW-complex
c) there are CW complexes which cannot be triangulated
3. Smooth manifold is PL so in particular can be triangulated and is CW complex
4. a) Any compact topological manifold of dimension $\leq 3$ can be triangulated
b) for any $n >3$ there are compact topological manifolds of dimension $n$ which can not be triangulated (known earlier for $n=4$, quite recent for $n>4$
5. a) Any compact manifold of dimension $ \neq 4$ is homeomorphic to CW complex
b) for $n=4$ this is an open problem
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truebaran
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Where do you think there are inconsistencies? – Emilio Ferrucci Aug 11 '14 at 21:42
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For example some people stated that 5 a) is valid only on the homotopic level – truebaran Aug 11 '14 at 22:50
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The proof that I know of (involving a handlebody decomposition, which uses Morse theory) shows that any smooth manifold (and in particular any manifold of dimension $\neq 4$ is smoothable) is homotopy equivalent to a $CW$ complex. I tried to do some digging (see for example this answer and the references provided in it), but in the end all the papers and books only seem to discuss the question of homotopy type. If I manage to find an answer I'll write it up later! – Emilio Ferrucci Aug 12 '14 at 09:19