I am working on the "A comprehensive introduction to differential geometry" by Michael Spivak. I have a question about an exercise in chap 2, on differential structures.
Let $M$ be a manifold, and $U$ and $V$ be open subsets of $M$. The functions $x:U\rightarrow x(U)\subset\mathbb{R}^n$ and $y:V\rightarrow y(V)\subset\mathbb{R}^n$ are co-ordinate mappings on $M$. Spivak then says two such maps are $C^{\infty}$-related if:
$y\circ x^{-1}:x(U\cap V)\rightarrow y(U\cap V)$
$x\circ y^{-1}:y(U\cap V)\rightarrow x(U\cap V)$
are both $C^{\infty}$
An exercise asks to show that this is not an equivalence relation. I can't see what is wrong with my reasoning, which seems to show that it is an equivalence relation. Here is my proof:
To check it is an equivalence relation we need to prove the 3 properties: Reflexivity, Symmetry, Transitivity. The relation is: $x\sim y$ if $x$ and $y$ are $C^{\infty}$-related
Reflexivity: $x\circ x^{-1} = Id$ which is a $C^{\infty}$ map.
Symmetry: If $x\sim y$ then $y\sim x$ since the $C^{\infty}$-related condition is symmetric.
Transitivity: If $x\sim y$ and $y\sim z$ then $x\circ y^{-1}$ and $y\circ z^{-1}$ are both $C^{\infty}$, so $(x\circ y^{-1})\circ (y\circ z^{-1})=x\circ z^{-1}$ is $C^{\infty}$. Similarly, $y\circ x^{-1}$ and $z\circ y^{-1}$ are both $C^{\infty}$, so $(z\circ y^{-1})\circ (y\circ x^{-1})=z\circ x^{-1}$ is $C^{\infty}$. So the transitive condition also holds. (am I wrong in assuming that the composition of $C^{\infty}$ maps is a $C^{\infty}$ map)
Thanks, Seb