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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.

Minkowski
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1 Answers1

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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:

  1. 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.

  1. 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.

Moishe Kohan
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    As a side note: this difference between open and closed manifolds is important for $n=4$. As Moishe pointed out in this MSE answer (https://math.stackexchange.com/a/2238453/386744), Quinn shows in that same paper that every open 4-manifold is smoothable and then has a CW structure (actually a triangulation). For closed 4-manifolds this is an open question. – Minkowski Jan 04 '22 at 17:51