Equivalence of atlases

Question

It is established that Equivalence of atlases is an equivalent relation. Now consider the real line $\mathbb{R}$ and the following one chart atlases $\mathcal{A} = \lbrace (\mathbb{R},Id)\rbrace$, $\mathcal{B} = \lbrace ( (0,1),Id^{2})\rbrace$ and $\mathcal{C} = \lbrace (\mathbb{R},Id^{3})\rbrace$ on $\mathbb{R}$. We can show that $\mathcal{A}$ are $\mathcal{B}$ equivalent since the chart in $\mathcal{A}$ and the chart in $\mathcal{B}$ are compatible and $\mathcal{B}$ and $\mathcal{C}$ are also equivalent since the chart in $\mathcal{B}$ and the chart in $\mathcal{C}$ are compatible. But $\mathcal{A}$ is not equivalent to $\mathcal{C}$ because $(\mathbb{R}, Id^{2})$ and $(\mathbb{R}, Id^{3})$ are not compatible. This is a counter example. Can any help?

Answer 1

(I suppose that $Id^3(x)=x^3$)

First, as said studiosus, is "meaningless to talk about equivalence of atlases on different spaces". Then, $\mathcal{A}$ and $\mathcal{C}$ are indeeed nonequivalent atlases but the intermediate comparisons are... meaningless. If the underlying space is $(0,1)$ then the atlases are equivalent.

Bonus fact: $(\Bbb R,\mathcal A)$ and $(\Bbb R,\mathcal C)$ are diffeomorphic. Try to find some diffeomorphism (Id isn't).