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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?

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  • $\begingroup$ Being a smooth manifold is a structure, not a property; you need to specify a smooth structure on a topological manifold (this is a property) and nothing in the definition of a CW complex offers such a thing. $\endgroup$ – Qiaochu Yuan Oct 10 '18 at 18:04
  • $\begingroup$ It assumed here that a smooth structure is given to the cell. I'll edit to make it clearer. $\endgroup$ – kaba Oct 10 '18 at 18:24
  • $\begingroup$ From where? Then of course the answer depends on which smooth structure was given, because there are exotic smooth structures on the open $k$-ball when $k = 4$: en.wikipedia.org/wiki/Exotic_R4 $\endgroup$ – Qiaochu Yuan Oct 10 '18 at 18:25
  • $\begingroup$ The exotic R4 answers the question of whether a smooth k-cell in isolation (i.e. a topological space homeomorphic to a $k$-ball and with smooth structure) is diffeomorphic to a ball (no). What still remains unclear is whether being a part of a CW complex could force a diffeomorphism. Essentially the question is whether it is possible to extend the exotic smooth manifold to a smooth manifold with boundary where the boundary consists of the cell's boundary-cells. $\endgroup$ – kaba Oct 10 '18 at 18:46
  • $\begingroup$ The exotic R4 page mentions that any smooth n-manifold which is homeomorphic to an n-ball is also diffeomorphic to an n-ball for $n \neq 4$. Therefore, it is possible that a small constraint such as being part of a CW-complex could remove the exception also for $n = 4$. $\endgroup$ – kaba Oct 10 '18 at 18:59
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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).

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  • $\begingroup$ Yes, I know the exotic sphere. But I was hoping that because the $k$-cell is part of a CW-complex, that would help to rule out such problems. $\endgroup$ – kaba Oct 10 '18 at 15:23
  • $\begingroup$ I don't think so, because there is no particular restriction on the homeomorphisms that "define" the cells in the definition of a CW-complex. Even more, you can give a CW-complex structure to an exotic sphere by taking a cell of the right dimension and the right homeomorphism. $\endgroup$ – Giuseppe Bargagnati Oct 10 '18 at 15:26
  • $\begingroup$ Each homeomorphism must extend in a continuous manner to the boundary cells, so that restricts them somewhat, right? The exotic sphere is homeomorphic to a sphere, and not a ball, so while it works as a cautionary tale, it does not seem to provide a counter-example. $\endgroup$ – kaba Oct 10 '18 at 15:41
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    $\begingroup$ Each (open) $k$-cell is by definition homeomorphic to an open $k$-ball. But what do you mean by the statement that a cell is a smooth $k$-manifold? Of course it can always be given a smooth structure, but I do not see any connection to the concept of CW-complex. Or do you mean a closed cell which can be given the structure of a smooth $k$-manifold with boundary? Also in that case it is a question about manifolds and not a question about CW-complexes. $\endgroup$ – Paul Frost Oct 10 '18 at 16:50
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    $\begingroup$ getting off topic, @QiaochuYuan perhaps the term is the "XY problem" $\endgroup$ – Alvin Jin Oct 10 '18 at 18:36

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