Some new engulfing theorems

Abstract

The technique of engulfing has become one of the most useful tools in the study of topological embeddings. In this paper, we extend the current engulfing theorems, especially the engulfing theorems. We view engulfing from Stallings' viewpoint [10]: consider an open set U in a manifold and determine conditions under which an isotopy of the manifold may be found such that the image of U at the end of the isotopy contains a given polyhedron. We also wish to require that the isotopy be fixed on parts of the polyhedron that were initially contained in U. (Zeeman [14] presents a different viewpoint of engulfing.) Radial engulfing, originally conceived by Connell [4], adds the restriction that the isotopy should only move in certain directions. ConnelΓs preferred directions were with respect to the origin in Euclidean space, hence the name radial engulfing. Bing [1] generalized ConnelΓs notion to other manifolds. Bing's engulfing theorem required that the dimension of the polyhedron to be engulfed be no greater than m - 4 where m is the dimension of the manifold (codimension 4). The main result of this paper (Theorem 2.1) is to prove Bing's theorem in codimension 3 and to slightly improve it in codimensions 4 and greater.