Embeddings in Generalized Manifolds

We prove that a (2m n+1)-connectedmap f : Mm ! Xn from a compact PL m-manifoldM to a generalized n-manifoldX with the disjoint disks property, 3m 2n 2, is homotopic to a tame embedding. There is also a controlled version of this result, as well as a version for noncompactM and proper maps f that are properly (2m n+ 1)-connected. The techniques developed lead to a general position result for arbitrary maps f : M ! X, 3m 2n 2, and a Whitney trick for separating PL submanifolds of X that have intersection number 0, analogous to the well-known results when X is a