Context:
- Continuous maps between topological spaces are structure preserving in the following sense:
Given two topological spaces $(X,\tau_X),(Y,\tau_Y)$ (where $(X,\tau_X)$ is the topological space of set $X$ endowed with a topology $\tau_X$; similarly $(Y,\tau_Y)$), a map $f:(X,\tau_X)\to (Y,\tau_Y)$ is continuous iff $f^{-1}(V)\subset X$ is open whenever $V\subset Y$ open.
- A smooth manifold is a Hausdorff second countable locally Euclidean space with a smooth structure. The smooth structure is one where you take a smooth atlas and consider the unique maximal smooth atlas generated by it.
Problem:
Question 1: What is the structure on a smooth manifold? Is it this maximal smooth atlas itself or merely the requirement that the co-ordinate charts (which the underlying topological manifold already has) need to be smoothly compatible?
Question 2: I want to show or think of smooth maps as the maps that preserve this structure of smooth manifolds. How can I do that?
Here is my (WRONG) guess: given a map $f:M\to N$ ($M,N$ smooth manifolds), $f$ is smooth iff for every smooth chart $(V,\psi)$ in the smooth structure of $N$, we get a unique smooth chart $(U,\phi)$ (via some kind of taking pre-images) in the smooth structure of $M$.
Best Answer
So first things first, every diffeomorphism is also a homeomorphism by definition, so diffeomorphisms automatically preserve every topological property about manifolds.
Since Lee has already given you an excellent answer, I will provide an alternative but equivalent view. Apologies if you know nothing about sheafs.
Now a smooth manifold also comes equipped with it's sheaf of smooth functions $C^\infty_M$, that is for each open set $U\subset M$ we take continuous functions $f:U\rightarrow \mathbb R$ such that in every coordinate chart $(V\subset U,\phi)$, $f\circ \phi^{-1}$ is a smooth function on an open subset of $\mathbb R^n$ . A continuous map $F:M\rightarrow N$ is then smooth if and only if for every open set $U\subset N$, we have that: \begin{align} F^\sharp:C_N^\infty(U)&\longrightarrow C^\infty_M(F^{-1}(U))\\ f&\longmapsto f\circ F \end{align} is a morphism of rings which commutes with restriction maps. In other words, we have that $F$ is smooth if and only if this prescription above defines a sheaf morphism: $$F^\sharp:C^\infty_N\longrightarrow F_*C^\infty_M$$ This all follows pretty much directly from the definition of a smooth map, just rephrased in fancier language. However, this has the added benefit of telling us that $F$ is a diffeomorphism if and only if $F$ is a homeomorphism and $F^\sharp$ is an isomorphism of sheaves.
In fact more is true, but I believe what I am about to say is hard to prove. Two manifolds $M$ and $N$ are diffeomorphic if and only if the ring of global smooth functions, $C^\infty(M)$, is isomorphic to $C^\infty(N)$. So in a sense diffeomorphisms can be seen as the smooth maps between manifolds which induce isomorphisms on the rings of global smooth functions.