Closed 4-manifolds have CW-complex?

Question

It looks like for compact 4-manifolds this question is open: When is a compact topological 4-manifold a CW complex?

How about if we just consider closed 4-manifolds, does that have an answer/make the question simpler?

Answer 1

Every closed 4-manifold is homeomorphic to a CW complex if and only if every compact 4-manifold with nonempty boundary is homeomorphic to a CW complex.

Answer 2

From Hatcher, p. 529:

"Corollary A.12. A compact manifold is homotopy equivalent to a CW complex.

One could ask more refined questions. For example, do all compact manifolds have CW complex structures, or even simplicial complex structures? Answers here are considerably harder to come by. Restricting attention to closed manifolds for simplicity, the present status of these questions is the following. For manifolds of dimensions less than 4, simplicial complex structures always exist. In dimension 4 there are closed manifolds that do not have simplicial complex structures, while the existence of CW structures is an open question. In dimensions greater than 4, CW structures always exist, but whether simplicial structures always exist is unknown, though it is known that there are n manifolds not having simplicial structures locally isomorphic to any linear simplicial subdivision of R n , for all n ≥ 4. For more on these questions, see [Kirby & Siebenmann 1977] and [Freedman & Quinn 1990]."