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

diffeomorphismmanifolds

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.

Best Answer

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 $0$.
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