Characterization of taming sets on 2-spheres

Abstract

1. Suppose that a 2-sphere S in E3 is tame modulo a closed subset F, and suppose that F is tame (i.e., F lies on a tame 2-sphere in E3). Is S tame? And if S is not tame, at which points of F can S be wild? These questions are answered by Theorem 1.1. The author is deeply indebted to C. E. Burgess, who in private conversation pointed out how Lemma 2.3 and linking arguments can be used to establish special cases of Theorem 1.1 (see [6]). If E is a positive number, let Fe denote the closed subset of F consisting of those points of F which lie in components of F of diameter equal to or greater than E. Let F# denote the subset of F which consists of those points of F which are degenerate components of F. The set F# is not necessarily closed.