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I'm looking for examples of smooth proper embeddings between connected compact manifolds of the same dimension that are not diffeomorphisms. I remember having seen an example with $S^7$ in mathoverflow, but now I can't find it.

EDIT: More generally, I'm looking for examples of smooth embeddings that are homeomorphisms but not diffeomorphisms. I remember having seen an example involving exotic $S^7$'s in mathoverflow, but now I can't find it.

EDIT: At first I erased the first question. Here it is again.

user17786
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2 Answers2

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(Here is an answer to the new version of the question.)

There are no such examples!
Indeed, since $f:M\to N$ is a homeomorphism, the manifolds $M$ and $N$ have the the same dimension by invariance of dimension ( a purely toplogical result).
But then the embedding $f$, being an immersion, is a local diffeomorphism which proves that $f^{-1}:N\to M$ is also smooth, so that finally $f$ must be a diffeomorphism.

Georges Elencwajg
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There are no such examples!
The embedding $f:M\to N$ is in particular an immersion and since $M$ and $N$ have the same dimension, it is a local diffeomorphism.
Thus $f$ is open, has an open image $f(M)$ and since $f(M)$ is compact (because $M$ is) we conclude by connexity of $N$ that $f(M)=N$.
Since an embedding is injective, $f$ is a smooth homeomorphism (remember that $f$ is open!).
Finally, since $f$ is a local diffeomorphism, the inverse map $f^{-1}:N\to M$ is smooth too and $f$ is a diffeomorphism.

Georges Elencwajg
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  • This is an answer to a previous version of the question. To see what it was, look at the edit history of the question ($M$ was assumed compact, the manifolds were assumed of the same dimension, $f$ was not assumed to be a homeomorphism)) – Georges Elencwajg Mar 05 '13 at 11:57