Suppose a $k$-cell in a CW complex is given a smooth structure, so that it is also a smooth $k$-manifold. Is that cell diffeomorphic to the open $k$-ball?
2026-03-25 06:09:06.1774418946
Smooth cell in a CW complex
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It depends on $k$...in a CW complex a (open/closed) cell is only homeomorphic to a open/closed ball; you can have different differentiable structures on a manifold that are homeomorphic but not diffeomorphic in some dimensions $\geq 4$ (see https://en.wikipedia.org/wiki/Exotic_sphere for a "famous" example).