Embeddings in generalized manifolds

We prove that a ($2m-n+1$)-connected map $f\colon M^m\to X^n$ from a compact PL $m$-manifold $M$ to a generalized $n$-manifold $X$ with the disjoint disks property, $3m\le 2n-2$, is homotopic to a tame embedding. There is also a controlled version of this result, as well as a version for noncompact $M$ and proper maps $f$ that are properly ($2m-n+1$)-connected. The techniques developed lead to a general position result for arbitrary maps $f\colon M\to X$, $3m\le 2n-2$, and a Whitney trick for separating $P\hspace*{-1pt}L$submanifolds of $X$ that have intersection number 0, analogous to the well-known results when $X$ is a manifold.