Splitting manifold approximate fibrations

Suppose M is a topological m-manifold, X is a generalized n-manifold satisfying the disjoint disks property (DDP), m>n⩾5, f : M→X is an approximate fibration, with fiber the shape of a closed topological manifold F, and Y is a closed, 1-LCC, codimension three subset of X. We examine conditions under which f is controlled homeomorphic to an approximate fibration g : M→X such that g|g−1(Y) : g−1(Y)→Y is, in some sense, an improvement of f|f−1(Y). One of the main results is that if Y is a generalized manifold, and if f|f−1(Y) : f−1(Y)→Y is fiberwise shape equivalent to a manifold approximate fibration p : E→Y, and Wh(π1(F)×Zk)=0, k=0,1,…, then f is controlled homeomorphic to a manifold approximation g : M→X such that g|g−1(Y) : g−1(Y)→Y controlled homeomorphic to p : E→Y.