Splitting Manifold Approximate Fibrations

Suppose M is a topological m-manifold, X is a generalized nmanifold 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 .