Some conditions for manifolds to be locally flat
C. Lacher · Transactions of the American Mathematical Society · 1967
Consider manifolds McJV 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 A/?" Without mentioning the case when N is three dimensional, this question has been considered before by Cantrell-Edwards 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.£" is euclidean «-space, £" is the closed unit ball in £", and S" is the one-point compactification of £".Sn is triangulated so that £n and £n inherit their triangulations from S\ When m<n, we identify £m with RmxO<=Rn.Thus we have Rm<=Rn<=Sn and Bm<=Bn^Sn for m<n.An n-cell (n-sphere, open n-cell) is a space homeomorphic to Bn (resp.Sn, resp.£n).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 which have open n-cell neighborhoods in N; the boundary of TV (denoted by Bd N) is the complement of AMnt N of Int N.Let M and N he manifolds of dimension m and n, respectively, with M<=lnt N. M is said to be locally flat in N at the point x e Int M if x has a neighborhood U in N such that (¿7, Un M)x(Rn, Rm); i.e., the pairs (U, Un M) and (£", Rm)