The Engulfing Theorem for Topological Manifolds

THEOREM. Every locally tame closed set X, of dimension p ? n 3 in a p-connected topological n-manifold-without-boundary M, can be engulfed by any (p 1)-connected open set V, such that V is compact.' This means that there is a self-homeomorphism h, of M such that X ' h V. There is an isotopy of h to the identity composed of a finite number of small pushes, each of which moves the points of a subset of a euclidean neighbourhood along parallel lines and leaves all else fixed. The theorem is in fact proved for a somewhat larger class of sets X, the p-dominated sets (p < n 3) defined in ? 9. By an argument due to E. H. Connell, based on Stallings' stretching process between dual skeletons, the Poincare hypothesis for topological n-manifolds (n ? 5) is derived from this form of the engulfing theorem.