Approximating homeomorphisms by diffeomorphisms on manifolds

Question

Let $M$ and $M'$ be smooth manifolds (compact if necessary)It is known that every continuous map $f:M\rightarrow M'$ can be approximated by smooth maps. What if $f$ is homeomorphism; when can we approximate it by diffeomorphism? What if $M$ and $M'$ are lie groups?Exmple: every homeomorphism between surfaces can be isotoped to a diffeomorphism

Answer 1

This would not work if you have two homeomorphic smooth manifolds which are not diffeomorphic, and there is a "well known" example:

Édit:you have the exotic spheres that work as counterexample

So you should reformulate you question with the additional condition: $M$ and $N$ are diffeomorphic, however you might have a result that only ensures local diffeomorphism instead of global.