Is every self-homeomorphism homotopic to a diffeomorphism?

Solution 1:

In dimensions 2 and 3 every homeomorphism is isotopic to a diffeomorphism (this should be in Moise's book "Geometric topology in dimensions 2 and 3", it also follows from Kirby and Siebenmann's work). In dimension 4 there are self-homeomorphisms of simply-connected smooth compact manifolds which are not homotopic to diffeomorphisms. This follows e.g. from invariance of the $\pm$ canonical class of smooth algebraic surfaces under diffeomorphisms, while, by Freedman's work, any automorphism of the intersection form is induced by a homeomorphism.

Edit. One more useful thing: The group of homeomorphisms of a topological manifold is locally contractible (with respect to the $C^0$ topology), this is a theorem by Chernavskii (1969). Thus, if you can approximate a homeomorphism by diffeomorphisms, they will be isotopic (for sufficiently close approximation).

Solution 2:

First, note that a homeomorphism $f$ of compact smooth manifolds is homotopic to a diffeomorphism if and only if one can approximate $f$ arbitrarily well by diffeomorphisms.

For $n \leq 3$, this paper of Munkres claims as a corollary that a homeomorphism $f: M \to N$ of smooth manifolds may be approximated arbitrarily well by a diffeomorphism. This settles my question here. There is a harder question of whether or not every homeomorphism is isotopic to a diffeomorphism. The consensus on this MathOverflow question and answers seem to be that it's true for $n=2$, though I haven't checked the stated references. Ian Agol's answer there sounds like it's true for $n=3$, but I'm not sure.

This note of Stefan Müller gives a proof that for $n \geq 5$, a homeomorphism of compact $n$-manifolds can be approximated arbitrarily well by diffeomorphisms if and only if the same homeomorphism is isotopic to a diffeomorphism.

Every related question in $n=4$ seems to be wide open.