TITLE: Making a bijection into a diffeomorphism
QUESTION [6 upvotes]: 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?

REPLY [4 votes]: 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.