A Generalization of Thom Classes and Characteristic Classes to Nonspherical Fibrations

Let X be a polyhedron, and let Fx denote the contravariant functor consisting of fiber homotopy types of Hurewicz fibrations over a given base whose fibers are homotopy equivalent to X. A fundamental theorem on fiber spaces states that Fx is a representable homotopy functor and a universal space for Fx is the classifying space for the topological monoid of self-equivalences of X [2; 5]. Frequently, algebraic topological information about the associated universal fibration yields information about arbitrary fibrations with fiber (homotopy equivalent to) X. However, present knowledge of the algebraic topological properties of the universal base space is extremely limited except in some special cases.