Uniform approximation of homeomorphisms by diffeomorphisms

2009/01/31 by Stefan Müller
Mathematics · #Diffeomorphism #Dimension (graph theory) #Discrete mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homeomorphism (graph theory) #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pure mathematics #Volume (thermodynamics) #math.DG #math.DS #msc:28D05 #msc:57Q55 #msc:57R12 #msc:58C35

paper · pdf · doi:10.1016/j.topol.2014.10.003

published as Topology Appl. 178 (2014) 315-319 · 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

arxiv created 2014/05/22 · openalex publication_date 2014/10/15 · arxiv updated 2016/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• C⁰-coerciveness of Moser's problem and smoothing area preserving homeomorphisms
• Structure of the group of homeomorphisms preserving a good measure on a compact manifold
• An application of gauge theory to four-dimensional topology
• An obstruction to smooth isotopy in dimension 4