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This is a very naive question. Every manifold (assumed to be paracompact) is a CW-complex?

Thanks.

Bilateral
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    Using Morse theory one can show that a smooth manifold is homotopic to a CW complex. – Seth Mar 27 '15 at 15:38
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    Better than that, @Seth, they all actually are CW complexes. (Instead of contracting the handlebody diagram down to the cores, just give the handles a workable CW structure; that or just pick a triangulation.) I think topological manifolds are all homotopy equivalent to CW complexes. If you google this question you should find some MO questions that help. –  Mar 27 '15 at 16:09
  • See http://math.stackexchange.com/questions/593041/cw-complexes-and-manifolds – archipelago Mar 28 '15 at 14:06

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