Approximate fibrations and bundle maps on Hilbert cube manifolds

This paper is concerned with parameterized families of approximate fibrations from a compact Hilbert cube manifold M to a compact polyhedron B. The main result shows how to straighten out certain of these families to be nearly like a product. As an application of this technique, it is shown that an approximate fibration p:M → B can be approximated arbitrarily closely by bundle maps if and only if p is homotopic via approximate fibrations to a bundle map. Another result is that the space of bundle maps from M to B is locally n-connected for each n ? 0.