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

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 $M$?

A reference would be helpful.

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    $\begingroup$ We know it's true when M=[0,1] : forums.futura-sciences.com/mathematiques-superieur/… $\endgroup$
    – Dattier
    Commented Dec 18, 2019 at 9:54
  • $\begingroup$ Maybe the following can also be interesting to you: P. Goldstein, P. Hajlasz, $C^1$ mappings in $R^5$ with derivative of rank at most 3 cannot be uniformly approximated by $C^2$ mappings with derivative of rank at most 3 J. Math. Anal. Appl. 468 (2018), 1108-1114. [arXiv] $\endgroup$
    – Behnam Esmayli
    Commented Dec 19, 2019 at 4:03

1 Answer 1

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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 $[\Sigma] = - [ \Sigma] \in \Theta_d$.

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    $\begingroup$ I suppose this is well-known to experts, but for the others: what would be an example of a homomorphism not isotopic to a diffeomorphism? $\endgroup$
    – abx
    Commented Dec 18, 2019 at 8:40
  • $\begingroup$ Thank you very much. Actually you have made two statements: are there references for both of them? $\endgroup$
    – asv
    Commented Dec 18, 2019 at 9:11
  • $\begingroup$ This is a great answer. Just one more question: is it hard to define $\Theta_d$? $\endgroup$
    – asv
    Commented Dec 18, 2019 at 10:13
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    $\begingroup$ @MKO: it's not hard to define. Take the oriented homotopy $n$-spheres, up to orientation-preserving diffeomorphism. They have a connect-sum operation, and that operation turns them into a group. Finding non-trivial elements in this group, especially ones that are not 2-torsion is quite a bit more work. Not being 2-torsion means the homotopy-sphere is not diffeomorphic to its orientation-reverse. $\endgroup$
    – Ryan Budney
    Commented Dec 19, 2019 at 8:35

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