Freedman's $E_8$manifold is nontriangulable, as proved on page (xvi) of the AkbulutMcCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3spheres". Kirby showed that a compact 4manifold has a handlebody structure if and only if it is smoothable: 1 and 2. When is a compact topological 4manifold a CW complex?
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