Let p : E → B be a locally
trivial fiber bundle between closed manifolds where dimE ≥ 5 and B has a
handlebody decomposition. A controlled homotopy topological structure (or a
controlled structure, for short) is a map f : M → E where M is a closed manifold of
the same dimension as E and f is a p−1(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 metriccontrol 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.
