Topologically, quasiconformally or Lipschitz locally flat approximation of embeddings

Let CAT denote either the category LIP of locally bi-Lipschitz embeddings or the category LQC of locally quasiconformal embeddings. The following theorem consists of representative special cases of our main results: For range manifolds not of dimension 4, in codimensions different from 2 a closed topological embedding between CAT manifolds without boundary can be approximated by locally CAT flat embeddings, these approximations being equivalent up to a small ambient CAT isotopy. The auxiliary analogous TOP results extend known results and are interesting in themselves.