Uniform approximation of homeomorphisms by diffeomorphisms

TL;DR

This paper establishes conditions under which homeomorphisms of smooth manifolds can be uniformly approximated by diffeomorphisms, highlighting the role of isotopy and volume preservation in higher dimensions.

Contribution

It proves that compactly supported homeomorphisms isotopic to diffeomorphisms can be approximated by diffeomorphisms, including volume-preserving ones, in dimensions five and higher.

Findings

Homeomorphisms isotopic to diffeomorphisms can be approximated by diffeomorphisms.

Volume-preserving homeomorphisms can be approximated by volume-preserving diffeomorphisms.

Approximation is possible in the uniform topology for compactly supported maps.

Abstract

We prove that a compactly supported homeomorphism of a smooth manifold of dimension greater or equal to 5 can be approximated uniformly by compactly supported diffeomorphisms if and only if it is isotopic to a diffeomorphism. If the given homeomorphism is in addition volume preserving, then it can be approximated uniformly by volume preserving diffeomorphisms.