Given a set $M$, one that can be made into a smooth manifold, and a bijection $f:M\to M$, does there exist a differentiable structure on $M$ such that $f$ is a diffeomorphism? In case it's not always true, what should $M$ and $f$ satisfy in order to make that possible? If such an structure does exist, is it unique in some sense, in general, are any two differentiable structures on $M$ sharing the disired property diffeomorphic to each other?
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2If "set" is replaced by "topological space" and "bijection" is replaced by "homeomorphism", the question is a classical one first answered in the negative by Bing in the 1950's by finding a topological involution of $S^3$ not topologically conjugate to any smooth involution. see http://mathoverflow.net/questions/204027/when-a-homeomorphism-is-a-diffeomorphism-w-r-t-to-a-suitable-smooth-structure – PVAL-inactive Jul 07 '15 at 16:58
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Proffering a counterexample:
If the set of non-fixed points of $f$ is non-empty and finite, then we cannot find such a differentiable structure. The set of fixed points of a diffeomorphism is always closed, but the complement of a finite non-empty set on a manifold is never closed.
I don't know what extra requirements would make this true.
Jyrki Lahtonen
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True, @PVAL. I thought about adding a clause that the set should be uncountable. Upvoting your comment will do, as this answer is not worth a bump for such a reason :-) – Jyrki Lahtonen Jul 07 '15 at 17:10