Controlled homotopy topological structures, Paci"c

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 ~ ι (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. 1. Introduction. In §