I have stated this question incorrectly before, sorry.
2026-04-29 00:56:15.1777424175
Do smooth compact connected manifolds admit CW compositions with a single 0-cell?
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I don't think it is true, what bout closed interval $[0,1]$, it is certainly a smooth manifold, and as you can prove it cannot have one $0-cell$.
This result is true for closed smooth manifold, because you can find a Morse function on it with one minimum, i.e a critical point of index $0$. And then it corresponds to the $0-cell$ of the CW-presentation of that manifold. (Now if you consider the dual presentation, then this actually proves that a closed smooth manifold has a CW-structure with exactly one $n-cell$, where $n=dim$ of the manifold).