TITLE: Is every self-homeomorphism homotopic to a diffeomorphism?
QUESTION [21 upvotes]: Given a smooth manifold $M$, is every homeomorphism $M \to M$ homotopic to a diffeomorphism? 
Hirsch's "Differential Topology" has a proof that every $C^1$ diffeomorphism of smooth manifolds is homotopic to a $C^\infty$ one, but as far as I can tell, says nothing about the case of $C^0$ automorphisms.
If false in general, is the above claim true in dimensions at most $3$? (If somehow false because of exotic smooth structures - is it true for topological manifolds supporting only one smooth structure?)

REPLY [8 votes]: In many dimensions, exotic spheres are counterexamples.
For example, the group of $7$ spheres is diffeomorphic to $\mathbb{Z}_{28}$.  Pick any element $\Sigma$ which doesn't not have order $2$, for example, a generator.

There is an orientation reversing homeomorphism $f:\Sigma\rightarrow \Sigma$ (so $f$ has degree $-1$), but there is no such orientation reversing diffeomorphism.

Since the degree is a homotopy invariant, this provides examples of what you want.
Proof:  Since $\Sigma$ is homeomorphic to $S^7$ and $S^7$ admits an orientation reversing homeomorphism, so does $\Sigma$.  Now, in the group of exotic spheres, the inverse element is the same sphere with orientation reversed.  Hence, if $\Sigma$ admits an orientation reversing diffeomorphism, then $\Sigma = -\Sigma \in \mathbb{Z}_{28}$, that is, the order of $\Sigma$ in $\mathbb{Z}_{28}$ is either $1$ or $2$.
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Of course, this proof carries over to any dimension for which the group of exotic spheres contains an element not of order $2$.  This has to occur if the order of the group is not a power of $2$, so, according to the chart on wikipedia, there are examples in dimension $7, 10, 11, 13, 15, 19, 20$.  See also the OEIS sequence to find more examples.