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Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms

Paweł Goldstein, Zofia Grochulska, Piotr Hajłasz

TL;DR

The paper addresses the problem of extending local diffeomorphisms defined on balls to global diffeomorphisms and extends the result to homeomorphisms and bi-Lipschitz homeomorphisms. It develops a general gluing framework (Theorem T1) based on Palais' trick, then extends it to less regular classes (Theorems T2, T8) by relying on the stable homeomorphism theorem and annulus theorem. It emphasizes that while the original Cerf-Palais extension is elementary, its extensions to non-smooth contexts require deep results in geometric topology. These results provide a versatile tool for patching local maps into global structure on manifolds, with implications for invariance under connected sums and for Lipschitz topology.

Abstract

Cerf and Palais independently proved a remarkable result about extending diffeomorphisms defined on smooth balls in a manifold to global diffeomorphisms of the manifold onto itself. We explain Palais' argument and show how to extend it to the class of homeomorphisms and bi-Lipschitz homeomorphisms. While Palais' argument is surprising, it is elementary and short. However, its extension to bi-Lipschitz homeomorphisms and homeomorphisms requires deep results: the stable homeomorphism and the annulus theorems.

Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms

TL;DR

The paper addresses the problem of extending local diffeomorphisms defined on balls to global diffeomorphisms and extends the result to homeomorphisms and bi-Lipschitz homeomorphisms. It develops a general gluing framework (Theorem T1) based on Palais' trick, then extends it to less regular classes (Theorems T2, T8) by relying on the stable homeomorphism theorem and annulus theorem. It emphasizes that while the original Cerf-Palais extension is elementary, its extensions to non-smooth contexts require deep results in geometric topology. These results provide a versatile tool for patching local maps into global structure on manifolds, with implications for invariance under connected sums and for Lipschitz topology.

Abstract

Cerf and Palais independently proved a remarkable result about extending diffeomorphisms defined on smooth balls in a manifold to global diffeomorphisms of the manifold onto itself. We explain Palais' argument and show how to extend it to the class of homeomorphisms and bi-Lipschitz homeomorphisms. While Palais' argument is surprising, it is elementary and short. However, its extension to bi-Lipschitz homeomorphisms and homeomorphisms requires deep results: the stable homeomorphism and the annulus theorems.
Paper Structure (5 sections, 21 theorems, 31 equations)

This paper contains 5 sections, 21 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Palais
  3. Proof of Theorem \ref{['T1']}
  4. Topological local linearization
  5. Proof of Theorems \ref{['T2']} and \ref{['T8']}

Key Result

Theorem 1.2

Let $\mathcal{M}^n$ be an $n$-dimensional connected and oriented manifold. Assume that $B\subset\mathcal{M}^n$ is diffeomorphic to $B^n(0,1)$ and let $F:B\to \mathcal{M}^n$ be an orientation preserving diffeomorphism onto its image, that can be extended to a diffeomorphism of a neighborhood of the c

Theorems & Definitions (43)

  • Theorem 1.2
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Corollary 1.10
  • Lemma 2.1
  • Lemma 2.2
  • ...and 33 more