Local Homotopy Properties of Topological Embeddings in Codimension Two

A b str act . Suppose N is an (n− 2)-manifold topologically embedded in an n-manifold M . Chapman and Quinn have shown that N is locally flat in M provided that the local homotopy groups of M − N at points of N are infinite cyclic and that all the higher local homotopy groups of M−N at points of N vanish. The main theorem in this paper asserts that it is only necessary to assume the homotopy conditions in dimensions below the middle dimension; i.e., if the local fundamental groups are good and if M −N is locally k-connected at points of N for 2 ≤ k < n/2, then N is locally flat.