Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms

TL;DR

This paper discusses extending local diffeomorphisms, homeomorphisms, and bi-Lipschitz mappings to global transformations on manifolds, highlighting elementary and deep topological results involved.

Contribution

It explains Palais' elementary argument for diffeomorphisms and extends it to homeomorphisms and bi-Lipschitz maps using advanced theorems.

Findings

Palais' argument is elementary and short.

Extension to bi-Lipschitz homeomorphisms relies on the annulus theorem.

Extension to homeomorphisms uses the stable homeomorphism theorem.

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.