Uniform approximation of homeomorphisms by diffeomorphisms

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.

v4: 5 pages; long overdue revision; clarified and improved statements, notation, and proofs. The main theorem may already be known, but I have not been able to find a precise reference. After talking to a number of topologists without getting a satisfactory answer, I decided to write up the proof myself. Comments and references welcome

References in corpus (2)

• An obstruction to smooth isotopy in dimension 4
• $C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphisms

Cited by in corpus (2)

• Differentiable approximation of continuous semialgebraic maps
• Universal Joint Approximation of Manifolds and Densities by Simple Injective Flows