I've been trying to construct a smooth diffeomorphism between non-smooth manifolds. Unfortunately I don't think I know enough manifolds well enough to find an example of this.
Mostly I've been looking at potential diffeomorphisms between $(\mathbb{R},A_1)$ and $(\mathbb{R},A_2)$, with $A_1$ and $A_2$ non-smooth atlases for $\mathbb{R}$. Is it possible to construct this? Or can it be proven that if two manifolds are smooth-diffeomorphic, then the manifolds themselves must be smooth?
On
So to say that a map $f : X \to Y$ is smooth, or is a diffeomorphism, there have to be smooth structures on $X$ and $Y$. That is, smooth structures are required in the definition of a smooth map.
Here are the relevent definitions:
A smooth atlas for a manifold $X$ is as an atlas such that for any two charts $(U_1, \phi_1)$ and $(U_2, \phi_2)$, the transition function $\phi_2 \circ \phi_1^{-1} : \phi_1(U_1 \cap U_2) \to \phi_2(U_1 \cap U_2)$ is has continuous partial derivatives of all orders. A smooth structure on $X$ is just a maximal smooth atlas for $X$.
A smooth manifold is just a manifold with a smooth structure.
We say that a function $f : X \to Y$ between smooth manifolds is smooth if for every $p \in X$, there is a smooth chart $(U, \phi)$ for $X$ with $p \in U$ and a smooth chart $(V, \psi)$ for $Y$ with $f(p) \in V$ such that $F(U) \subseteq V$ and the transition function $\psi \circ f \circ \phi^{-1}$ is has continuous partial derivatives of all orders. Of course, by "smooth chart," we mean a chart in the smooth atlas.
The word "(smooth) diffeomorphism" is meaningless unless you're talking about smooth manifolds. If all you have is a pair of topological manifolds $M$ and $N$, it is not meaningful to say a map $f:M\to N$ is a diffeomorphism. To say $f$ is a diffeomorphism, you need to be able to say that $f$ and $f^{-1}$ are smooth, but you can't define this without a smooth structure on $M$ and $N$. It's like trying to define a ring-homomorphism between two groups.