Is every open topological $n$-manifold homeomorphic to a CW complex? $(n \geq 5)$
The question of whether a closed topological manifold admits a CW structure is well-understood (with positive answer) for $n \neq 4$, thanks to the work Quinn ($n=5$) and Kirby-Siebenmann $(n >5)$.
However I cannot find anything about open (non-compact without boundary) manifolds of dimension $n\geq 5$. At least for $n \leq 4$ the answer is known to be yes (trivial for $n \leq 3$ and work of Quinn for $n=4$, see here.
I do not understand why people restrict to closed manifolds in their MSE/MO answers. If you look at the original papers, you will find no such restrictions. Moreover, the result (existence of a handle decomposition) also holds for manifolds with boundary and, furthermore, its proven that one can extend the given handle decomposition on a (locally flat) submanifold to the entire manifold:
- In dimension $\ge 6$ this is in the book by Kirby and Siebenmann, page 104, Theorem 2.1:
Kirby, Robion C.; Siebenmann, Laurence C., Foundational essays on topological manifolds, smoothings and triangulations, Annals of Mathematics Studies, 88. Princeton, N.J.: Princeton University Press and University of Tokyo Press. V, 355 p. (1977). ZBL0361.57004.
- In dimension 5, this is due to Quinn, Theorem 2.3.1 in
Quinn, Frank, Ends of maps. III: Dimensions 4 and 5, J. Differ. Geom. 17, 503-521 (1982). ZBL0533.57009.