Is a diffeomorphism between two manifolds just a diffeomorphism between the sets of the manifolds

Question

I'm reading Intro to GR by Sean Carroll, and in that book he defines a $C^\infty$ n-dimensional manifold as a set M along with a maximal atlas. In the same chapter, he also defines a diffeomorphism between two sets $M$ and $N$ as a $C^\infty$ map $\phi: M \to N$ and a $C^\infty$ inverse $\phi^{-1}: N \to M$.

My question is, if there are two manifolds $A$ and $B$, which are comprised of sets $M$ and $N$ respectively, along with appropriate atlases, is a diffeomorphism between those two manifolds simply a diffeomorphism between the sets $M$ and $N$, or is it defined some other way? In an exercise at the end of the chapter, he asks whether we can make $\mathbb R^1 $ look like $\mathbb R^2$ by "clever choice of coordinate charts", which would seem to imply that choice of charts plays a role in whether a diffeomorphism can be found between two manifolds or not.

I am not experienced in the area of manifolds, so if I have made any mistakes or assumptions in asking this question, please let me know.

Answer 1

The manifold structure depends entirely on the choice of atlas. For some implicit manifolds it is really necessary to have a proper and well define atlas, but usually they are implicit (e.g. if you consider an open subset of $\mathbb{R}^n$)

However you must be careful, a topological space (not just a set) can have different manifold structures, here is an example:

• Take $R_0 = \mathbb{R}$ with canonical manifold structure.
• Take on the other side $R_1 = \mathbb{R}$ with the atlas given by the chart $x \in \mathbb{R} \mapsto x^3$ which is a homeomorphism.

You then have two different manifolds and you can check that $x \in R_{0} \mapsto \sqrt[3]{x} \in R_1$ is a diffeomorphism! It would not be if you considered it from $R_0$ to $R_0$.

To sum up the structure is very important but often implicit. To come back to your question, a nicer formulation would be "is there two manifold structures on $\mathbb{R}$ and $\mathbb{R}^2$ such that both are diffeomorphic". This question is equivalent to: is there an homeomorphism between both topological spaces, and the answer is no (this is not too hard to verify).

In summary, sets have no structure, topological space have a somewhat richer structure and manifolds an even richer one.

Edit:

• By canonical, I indeed mean $\operatorname{id}$ as chart (as usual for open subsets of $\mathbb{R}^n$.
• Notice that $\sqrt[3]{x}$ is a diffeo iff $\phi: x \mapsto (\sqrt[3]{\operatorname{id}(x)})^3 = x$ is (and it is).
• For $R_0$ to $R_0$ you'll notice that this function is not differentiable in $0$.