I'm a beginner of a smooth manifold.
I can not understand the difference of homomorphism and diffeomorphism between two smooth manifolds.
Let $M$ and $N$ be smooth manifolds. If $M$ and $N$ are homomorphic, then there exists a homomorphism $F:M\to N$.
I think $F$ is also a diffeomorphism between $M$ and $N$ because, for a chart $(U,\phi)$ of $M$, a pair $(F(U), \phi\circ F^{-1})$ becomes a chart of $N$ and a composite function $(\phi\circ F^{-1})\circ F \circ \phi^{-1}=\mathrm{Id}$ is smooth over $\phi(U)$.
Is there anything wrong with the above discussion?
Best Answer
Your confusion is between homeomorphisms and diffeomorphisms between smooth manifolds. Smooth manifolds are topological spaces with extra structure, which allows us to talk about smooth functions (not just continuous) on our spaces. Two topological spaces are 'the same' (in the sense that they possess the same topological properties) if they are homeomorphic, while two smooth manifolds are so if they are diffeomorphic.
Since smooth manifolds are very special topological space, we should expect 'equivalent' smooth manifolds, that is, diffeomorphic ones, to also be topologically equivalent. This is so: this is proposition 2.15c in John M. Lee's 'Introduction to Smooth Manifolds.'
But homeomorphic smooth manifolds are not neccesarily diffeomorphic: there exist different smooth structures on $S^7$, which make it into truly different, non-diffeomorphic smooth manifolds, although these different smooth manfiolds are homeomorphic to each other. See the wiki page on exotic spheres for more information.
Finally, to address the flaw in your argument, the chart which you define on $N$ doesn't have to be smoothly compatible with the atlas on $N$.