Let $M$ be a topological manifold. I can equip $M$ with a smooth atlas $\mathscr{A}$ and a smooth atlas $\mathscr{B}$. If there exists a diffeomorphism $f$ between the two resulting smooth manifolds, it doesn‘t necessarily follow that the identity is a diffeomorphism. (This would imply that $\mathscr{A}\cup\mathscr{B}$ is a smooth atlas and I know counterexamples.)
But the composition of two diffeomorphisms is again a diffeomorphism and we have $\;\text{id}_M=f^{-1}\circ f$. Where am I thinking wrong?
Best Answer
If $f : (M, \mathscr{A}) \to (M, \mathscr{B})$ is a diffeomorphism, then $f^{-1} : (M, \mathscr{B}) \to (M, \mathscr{A})$ is a diffeomorphism and hence $\operatorname{id}_M = f^{-1}\circ f : (M, \mathscr{A}) \to (M, \mathscr{A})$ is a diffeomorphism. However, that tells us nothing about whether $\operatorname{id}_M : (M, \mathscr{A}) \to (M, \mathscr{B})$ is a diffeomorphism or not.