Suppose that p is a fibration with total space E over a connected, finite CW complex B whose fibers are homotopy equivalent to a finite CW complex F. Then there is a fibration associated to p whose fibers are all homotopy equivalent to A(F), the algebraic K-theory space of F. Dwyer, Weiss, and Williams have constructed a section of this fibration, called the parametrized Euler characteristic of p, which has a lift to a parametrized excisive Euler characteristic if and only if p is fiber homotopy equivalent to a topological fiber bundle whose fibers are homeomorphic to a compact topological manifold, possibly with boundary.
Assuming that p does admit at least one compact topological fiber bundle structure, we can also try to classify the space of all such structures on p. We show that this space of structures on p is homotopy equivalent to the product of the space of all lifts of the parametrized Euler characteristic of p to a parametrized excisive Euler characteristic with the space of stable Euclidean bundles over E.