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Let $M$ be a smooth manifold. Is it always true that the group of diffeomorphisms is strictly contained in the group of homeomorphisms?

(I know this is true for $\mathbb{R}^n$, but that is only a local result when considering general manifolds).

Also, can it happen that $\text{Diff}(M)$ is of finite index in $\text{Homeo}(M)$? of countable index?

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    $\begingroup$ It is not of finite index. You can prove this yourself using the idea of Mark Joshi's answer. It's not even of countable index. Try proving this for $\text{Homeo}(\mathbb R)$ first for the idea. $\endgroup$
    – user98602
    Jul 3, 2015 at 16:24

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well every manifold will have a part diffeomorphic to the unit ball so all we have to do is construct a homeomorphism of the unit ball that is not a diffeomorphism and is the identity close to its boundary. This map can then be extended by the identity to the whole manifold.

For the unit ball, just take a cube root in each coordinate and then adjust away from the origin to get it to be the identity near the boundary.

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