the proper context for the "s"-cobordism theorem is the classifying space of "h"-cobordisms. if "m" is a cat manifold, then "h"("m") is a space that classifies bundles of "h"-cobordisms on "m". the "s"-cobordism theorem can be reinterpreted as the statement that the set of connected components of this space is the whitehead group of π("m"). this space contains strictly more information than the whitehead group; for example, the connected component of the trivial cobordism describes the possible cylinders on "m" and in particular is the obstruction to the uniqueness of a homotopy between a manifold and . consideration of these questions led waldhausen to introduced his "algebraic "k"-theory of spaces". the algebraic "k"-theory of "m" is a space "a"("m") which is defined so that it plays essentially the same role for higher "k"-groups as "k"(zπ("m")) does for "m". in particular, waldhausen showed that there is a map from "a"("m") to a space wh("m") which generalizes the map and whose homotopy fiber is a homology theory.