Let $M$ be a smooth closed manifold. Let $f\colon M\to M$ be a homeomorphism.



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> Does there exist a sequence of diffeomorphisms $f\_i\colon M\to M$ which conveges to $f$ uniformly, i.e. in $C^0$-topology:
>  $$\sup\_{x\in M}dist(f(x),f\_i(x))\to 0,\, \sup\_{x\in M}dist(f^{-1}(x),f^{-1}\_i(x))\to 0 \mbox{ as } i\to\infty,$$
>  where the distance $dist$ is taken with respect to a Riemannian metric on $M$?
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A reference would be helpful.



Approximation of homeomorphism by diffeomorphism
No. The space of homeomorphisms of a compact manifold is locally contractible:


A. V. Černavskiı̆. Local contractibility of the group of homeomorphisms of a manifold. Mat.
Sb. (N.S.), 79 (121):307–356, 1969.


So if there were such a sequence then for large enough $i$ the diffeomorphism $f\_i$ would be topologically isotopic to $f$. But there are homomorphisms which are not isotopic to diffeomorphisms. 


For example, let $\Sigma$ be a smooth homotopy $d$-sphere which does not have order 2 in the group $\Theta\_d$ of such (e.g. Milnor's exotic 7-sphere). As $\Sigma$ is homeomorphic to $S^d$ (by the topological Poincare conjecture) it admits an orientation-reversing homeomorphism $f : \Sigma \to \Sigma$. But this $f$ cannot even be homotopic to a diffeomorphism, for if it were it would mean that $[\Sigma] = - [ \Sigma] \in \Theta\_d$.


