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Is it always possible to put a cell structure on a manifold? in other words is it possible to decompose a manifold as a CW complex? I know that by Morse theory we always have a handle decomposition of the manifold which shows that it is homotopy equivalent to a CW complex, but it does not show that it is homeomorphic or diffeomorphic to a CW complex.

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    Apparently Kirby and Siebenmann (http://mathoverflow.net/questions/36838/are-non-pl-manifolds-cw-complexes) showed that every compact (topological) manifold is homeomorphic to a CW complex. "Diffeomorphic to a CW complex" doesn't make sense. Morse theory can be used to show that compact smooth manifolds are homeomorphic to CW complexes but the proof is apparently harder (http://mathoverflow.net/questions/86610/the-difference-between-a-handle-decomposition-and-a-cw-decomposition). – Qiaochu Yuan Jul 15 '14 at 19:22
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    Only in dimensions different from 4. In dimension 4 it is an open problem. I think, this questions was already asked at MSE. – Moishe Kohan Jul 15 '14 at 19:24

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