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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?

Fred Wealthman
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    Not sure what your notation means, but at a guess it seems to me like you're talking about the identity map (on different spaces) there, which, in particular, has the property that $Id^3=Id$. So... aren't $\mathcal A$ and $\mathcal C$ literally the same "atlas"? – Dustan Levenstein Mar 11 '14 at 17:30
  • It is also meaningless to talk about equivalence of atlases on different spaces. Maybe you mean "diffeomorphic"? – Moishe Kohan Mar 12 '14 at 02:28
  • I've edited ma question. You can go through it and See whether you can help. Thank You – Fred Wealthman Mar 12 '14 at 10:07
  • You still need to explain what you mean by $id^2$ and $id^3$; also, you haven't answered studiosus's comment. – Michael Weiss Mar 15 '14 at 03:11
  • I was ask to show that the compatibility of charts is not equivalence relation but the equivalence of atlases is an equivalence relation. so I used the charts for the in $\mathcal{A}$, $\mathcal{B}$ and the one in $\mathcal{C}$ as countable examples because the chart in $\mathcal{A}$ and the chart in $\mathcal{B}$ are compatible. Also the chart in $\mathcal{B}$ and the chart in $\mathcal{C}$ but the one in $\mathcal{A}$ and the one in $\mathcal{C}$ are not compatible.so I was thinking that you could use these as one chart atlases and use it as a counter example for the equivalence of atlases. – Fred Wealthman Mar 17 '14 at 12:47

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(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).

Martín-Blas Pérez Pinilla
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