Obstructions to stably fibering manifolds

Is a given map between compact topological manifolds homotopic to the projection map of a fiber bundle? In this paper obstructions to this question are introduced with values in higher algebraic K-theory. Their vanishing implies that the given map fibers stably. The methods also provide results for the corresponding uniqueness question; moreover they apply to the fibering of Hilbert cube manifolds, generalizing results by Chapman-Ferry.