Suppose $M$ is a compact connected manifold-with-boundary with non-empty boundary. What can be said on whether $M$ can be given a CW complex structure?
A similar problem has been discussed for compact connected manifolds (i.e. with empty boundary) here, where it is true for $\dim \neq 4$ and not known for $\dim = 4$. Suppose the case for $\dim = 4$ is also shown true someday. Can the case with non-empty boundary be reduced to the case with empty boundary; i.e. are these problems equivalent?