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Unfortunately, googling this question leads to conflicting answers. According to this source, the identity map on any smooth manifold is a diffeomorphism, but it's not according to this. I appreciate it if someone gave a definitive answer.

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    The statement that the questioner is referring to begins at the bottom of page 38, and continues onto page 39, ending with "Thus the identity map is not a diffeomorphism."2012-10-15
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    @Smith: the second link you found refers to a map which is the identity on underlying sets, but uses a different smooth structure in the source and the target. Personally, I think this is a bad abuse of notation. It does not refer to the identity map from a smooth manifold to itself considering only a single smooth structure, which is a diffeomorphism.2012-10-15

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