Some conditions for manifolds to be locally flat

Abstract

Consider manifolds Mc N and a subset X of M, and assume that M- X and X are locally nice in N. The general question considered in this paper is What conditions on X imply that M is nice in N? Without mentioning the case when N is three dimensional, this question has been considered before by CantrellEdwards in [9], by Cantrell in [6] and [7], by Edwards in [11], by Bryant in [5], by Lacher in [14], and elsewhere. However, in each of the above references the author restricts himself by either assuming that M lies in the trivial range or by assuming that X is a single point. The conditions derived in this paper make no dimensional restriction on M and assume only that Xx [0, 1] lies in the trivial range. The first three sections are devoted to studying embeddings of polyhedra into a manifold (in the trivial range). The polyhedra are allowed to intersect the boundary of the manifold. The results on embeddings in the trivial range constitute a major step in the proof of the main result of this paper (Theorem 4.2). The fourth and fifth sections derive some conditions for M to be nice in N when X lies in the boundary of M. The last section extends these results to the case when X lies in the interior of M, modulo a certain conjecture. 0. Definitions and notations. Rn is euclidean n-space, Bn is the closed unit ball in Rn, and Sn is the one-point compactification of Rn. Sn is triangulated so that Rn and Bn inherit their triangulations from Sn. When m<n, we identify Rm with RmxOczRn. Thus we have Rmc:Rnc:Sn and Bmc:Bnc:Sn for m<n. An n-cell (n-sphere, open n-cell) is a space homeomorphic to Bn (resp. Sn, resp. Rn). An n-manifold is a space N such that each point of N has a neighborhood whose closure is an n-cell; the interior of N (denoted by Int N) is the set of points of N