It is well know that a topological manifold $M$ is homotopy equivalent to a CW-complex $X$. In addition, if $M$ is compact, one even has $M \simeq X$ for some finite CW-complex.
In a Mathoverflow thread, it was discussed if one can take an $n$-dimensional CW-complex $X$ if $M$ is of dimension $n$. However, I have not found a clear reference in this thread, but I think most people believe it to be true.
My question is: Can we at least say for sure (with a good reference/reason) that we can find a finite dimensional CW-complex $X$ with $M \simeq X$ for every topological manifold $M$?
It seems as if Milnor in On spaces having the homotopy type of a CW-complex proves this claim, because the manifold gets embedded into a finite dimensional simplicial complex. But I have some trouble reading this article. Is there a better reference or resource for this fact? It seems quite useful to me.