Controlled Homotopy Topological Structures

Let p : E —> B be a locally trivial fiber bundle between closed manifolds where dim E > 5 and B has a handlebody decomposition. A controlled homotopy topological structure (or a controlled structure^ for short) is a map / : M —> E where M is a closed manifold of the same dimension as E and / is a p~ (ε)-equivalence for every ε > 0 (see §2). It is the purpose of this paper to develop an obstruction theory which answers the question: when is f homotopic to a homeomorphism, with arbitrarily small metric control measured in B? This theory originated with an idea of W. C. Hsiang that a controlled structure gives rise to a cross-section of a certain bundle over B, associated to the Whitney sum of p : E —• B and the tangent bundle of B.