An infinite-dimensional Schoenflies theorem

Abstract

Although important high-dimensional cases of the polyhedral Schoenflies theorem are yet unsettled (see [4] and [11]), there has been considerable success in the past ten years in generalizing the theorem, proved by Schoenflies in 1906, to the effect that every one-dimensional sphere embedded in the plane is tame. Of course Alexander's horned sphere and others show the necessity of a further condition to obtain a generalization to embeddings of S in E +1 for n> 1. Alexander proved tameness in the combinatorial and differential categories for n=2, M. H. A. Newman extended the result to include all finite dimensions in the combinatorial category [10] and Barry Mazur accomplished the same for the differential category [7]. In the topological category, Marston Morse showed that Mazur's result implies tameness of semilocally tame (bicollared) n-spheres in En+, [9] and at the same time Morton Brown gave an elegant proof independent of Mazur's result [3]. As might be expected, in infinite-dimensional normed linear spaces even the counterpart of the Jordan curve theorem does not hold. However, it is shown in this paper that with the bicollared condition one does get Jordan separation. In fact tameness is proved by a modification of Brown's technique. Hopefully this infinite-dimensional Schoenflies theorem will be as useful in infinite-dimensional topology as has been the case in finite dimensions. To give substance to this hope a number of applications are given including an annulus theorem, a contribution to the classification of convex sets and partial answers to questions raised by R. H. Bing and R. D. Anderson [2]. 1.0 DEFINITIONS. Unless otherwise stated, X denotes a topological space and E (with or without subscripts or superscripts) an infinite-dimensional normed linear space with the norm topology. Let Bt={xeFII xI ? t} and St={xeFII xI=t} for t > 0. If A c X, Bd A denotes the point-set boundary of A and 'A its interior. If h: (B2, S2) -> (A, Bd A) is a homeomorphism, then A is called a (closed) E-cell in X, h(B1) is a collared E-cell in X, collared by h and A is a tame E-cell if X= E and h extends to a space homeomorphism. On the other hand, if h: (B3 -B1, S3 u S1) -(A, Bd A) is a homeomorphism, A is called a (closed) E-annulus in X and h(S2) a bicollared E-sphere in X, bicollared by h. The prefix Ewill be omitted if no confusion is likely.