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?)
Best Answer
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).