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?

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 CWcomplexes as being a tool for describing homotopytypes rather than homeomorphism types, so my answer was to a weaker question than the one asked. $\endgroup$– Ryan BudneyAug 22, 2011 at 19:09

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 RanickiAug 22, 2011 at 21:56
1 Answer
Hatcher, Algebraic Topology, Corolary A.9
Every compact manifold, with or without boundary, is an E[uclidean]N[eighborhood]R[etract].
See also the following three numbered conclusions.
Or, in simpler terms, CW is a lot more flexible than, say, PL.
By "compact manifold", I expect you mean, in particular, an object $X$ with a finite very good cover  where, in case I need to be so careful, by "very" good cover, I mean that the boundary of every intersection of the Čech complex is a sphere. Without Loss Of Generality, every compact manifold has such a cover.
So fix one finite good cover, and refine it to a finite good cover by closed sets. Construe the Čech complex of this good finite closed cover as a simplicialsetover $X$. Its realization has the homotopy type of both $X$ (it is fiberwise finite contractible) and a CW complex: roughly, each open patch of the Čech complex is a cone over a sphere, and there are finitely many, so we are iteratively gluing in a finite family of products $\triangle^n \times C(\mathbb{S}^k)$. This doesn't literally produce a CW complex, in that the attaching of cells isn't in cellular order; however it is a finite process, amenable to iterative correction, and the Reader can supply the details as needed.
Since you mention Handlebodies I wonder: are you actually trying to get at which $4$manifolds are homeomorphically 4dimensional CWcomplexes?

1

4$\begingroup$ A compact 4manifold is a compact topological space which is locally homeomorphic to R^4. Borsuk proved in the 1930's that compact topological manifolds (in all dimensions) are ENR's. But without differentiable structure, it is by no means clear that a topological manifold has a good cover. And yes, I do mean to get at compact 4manifolds which are homeomorphic to (and so are) 4dimensional CW complexes. $\endgroup$ Aug 12, 2015 at 6:23