Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres". Kirby showed that a compact 4-manifold has a handlebody structure if and only if it is smoothable: 1 and 2. When is a compact topological 4-manifold a CW complex?
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asked Aug 22, 2011 at 18:32
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2$\begingroup$ I take it this is an open problem? Small note, but your question was effectively asked in another form, here: mathoverflow.net/questions/36838/… I usually think of CW-complexes as being a tool for describing homotopy-types rather than homeomorphism types, so my answer was to a weaker question than the one asked. $\endgroup$– Ryan BudneyCommented Aug 22, 2011 at 19:09
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1$\begingroup$ I have good reason to believe that it is an open question! Apologies - I hadn't seen the earlier posting mathoverflow.net/questions/36838/… $\endgroup$– Andrew RanickiCommented Aug 22, 2011 at 21:56
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