I'm reviewing some concepts diffferential manifolds and came across the exotic $R^4$ phenomenon: there are infinitely many non-diffeomorphic differential structures on $\mathbb{R}^4$. And it is pointed out that it is more easy to find non-diffeomorphic homeomorphisms than to find non-diffeomorphic manifolds. I do not understand what is it like for a homeomorphism to be non-diffeomorphism. I saw some answers saying that there is such an example even in $S^1$.
So I have a 2-part question:
- Can you give me an example in 1, 2, 3 dimensions, respectively? One such that it is homeomorphism but not diffeomorphism.
- Intuitively, what's the diffenrence between them? (Please don't tell me the definitions, I am very familiar with them. It's the geometric intuition I'm forgetting now.)
Here is an example of a homeomorphism that's not a diffeomorphism: $f(x)= x^{\frac{1}{3}}$. The reason it fails to be a diffeomorphism is that it's not differentiable at $x=0$. A related example is the function $g(x)= x^3$. This fails to be a diffeomorphism not because of the failure of differentiating. The function is a polynomial so its differentiable everywhere but it doesn't have a differentiable inverse. Its inverse is f(x)!
Geometrically what's happening is there is some point in which the derivative vanishes (or not defined). Both of which lead to the failure of being a diffeomorphism. Further, as the examples above illustrate, this only has to happen at a single point.