Singular regular neighborhoods and local flatness in codimension one

Abstract

For an ( n − 1 ) (n - 1) -manifold S S topologically embedded as a closed subset of an n n -manifold N N , we define what it means for S S to have a singular regular neighborhood in N N . The principal result demonstrates that S S has a singular regular neighborhood in N N if and only if the homotopy theoretic condition holds that N − S N - S is locally simply connected ( 1 1 -LC) at each point of S S . Consequently, S S has a singular regular neighborhood in N N if and only if S S is locally flatly embedded ( n ≠ 4 ) (n \ne 4) .