Homotopy type of a CW complex

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The only dimension in which not every compact manifold is homeomorphic to a CW complex is $4$. Does every such manifold have the homotopy type of a CW complex?

asked Jun 19, 2015 at 20:56

user74319

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$\begingroup$ Yes mathoverflow.net/questions/156266/… $\endgroup$
– Fernando Muro
Commented Jun 19, 2015 at 21:42
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$\begingroup$ Yes, see Corollary A.12 of Hatcher's book. $\endgroup$
– Andy Putman
Commented Jun 19, 2015 at 21:44

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