Lipschitz and quasiconformal approximation of homeomorphism pairs

Let CAT denote either the category LIP of locally bi-Lipschitz embeddings or the category LQC of locally quasiconformal embeddings. We prove that homeomorphisms between locally CAT flat CAT manifold pairs of arbitrary codimension can be approximated by CAT homeomorphisms, at least if there are no induced 4-submanifolds. It follows that a locally flat topological manifold pair satisfying the same dimensional restrictions admits a locally CAT flat CAT manifold pair structure. In the case of empty submanifolds these results are due to Sullivan (no boundaries) and Tukia and Vنisنlن (boundaries allowed).