Bundle theories for topological manifolds

Source abstract

Manifold approximate fibrations arise in the geometric topology of manifolds and group actions on topological manifolds. The primary purpose of this paper is to classify manifold approximate fibrations in terms of the lifting problem for a certain bundle. Our classification meshes well with the classical classifications of fibrations and bundles and, hence, we are able to attack questions such as the following. When is a fibration controlled homotopy equivalent to a manifold approximate fibration? When is a manifold approximate fibration controlled homeomorphic to a bundle?