A uniform generalized Schoenflies theorem

Abstract

The generalized Schoenflies theorem of M. Brown [2], [3] can be restated in the following way: If S~ is the equator of S, then any locally flat embedding ƒ : S~—>S can be extended to a homeomorphism Ft S-»S . The purpose of this paper is to show that, if n^ 5, the extension F can be constructed in a controlled manner; in particular, if ƒ:S~^»5 is close to the inclusion embedding, then F: S-*+S can be chosen to be close to the identity homeomorphism. Consequently if, ƒ, g :S'-^S are locally flat embeddings, n^ 5, and ƒ is close to g, then there is a homeomorphism H: S—^S which is close to the identity such that