"Unlike" non-diffeomorphic homeomorphisms

Question

We know that homeomorphic (topological isomorphisms) manifolds are not necessarily diffeomorphic (smooth isomorphisms).

We know that diffeomorphic (smooth isomorphisms) manifolds are necessarily homeomorphic (topological isomorphisms).

However Wikipedia had a statement:

Unlike non-diffeomorphic homeomorphisms, it is relatively difficult to find a pair of homeomorphic manifolds that are not diffeomorphic.

What is the emphasis on this "Unlike" on this "Unlike non-diffeomorphic homeomorphisms"? I do not know the logic of this emphasis?

It seems that according to @Eric Wofsey,

(1) "it is very easy to find an example of a homeomoprhism that is not a diffeomorphism."

(2) "it is relatively difficult to find a pair of homeomorphic manifolds that are not diffeomorphic."

Could you write an answer and explain why?

Answer 1

1. It is very easy to give examples of homeomorphisms between smooth manifolds which are not diffeomorphisms. For instance, the map $f: {\mathbb R}\to {\mathbb R}$ given by the formula $f(x)=x^3$ is a homeomorphism but is not a diffeomorphism. To see that $f$ is not a diffeomorphism, note that $f'(0)=0$, while a diffeomorphism has to have invertible derivative at each point. Alternatively, $f^{-1}(y)=y^{1/3}$ which is not differentiable at $0$. I leave it to you to verify that $f$ is a homeomorphism, you just need to check that both $f(x)=x^3$ and $f^{-1}(y)=y^{1/3}$ are continuous functions.

2. In contrast, it is hard to give examples (and even harder to prove) of two smooth manifolds which are homeomorphic but not diffeomorphic. In the known examples, it is not even obvious that two manifolds are homeomorphic to each other (this usually relies upon some theorems and the existence of a homeomorphism is not proven by writing down a formula). Once you prove that two manifolds are homeomorphic, you are then faced with the challenge that there is not diffeomorphism between the manifolds. The fact that some homeomorphism between two manifolds is not a diffeomorphism does not mean that there is no other map which will be a diffeomorphism. Look at the example in Part 1: In fact, the identity map is a diffeomorphism from ${\mathbb R}$ to itself.