whitehead torsion was eventually reinterpreted in a more directly "k"-theoretic way. this reinterpretation happened through the study of "h"-cobordisms. two "n"-dimensional manifolds "m" and "n" are "h"-cobordant if there exists an -dimensional manifold with boundary "w" whose boundary is the disjoint union of "m" and "n" and for which the inclusions of "m" and "n" into "w" are homotopy equivalences (in the categories top, pl, or diff). stephen smale's "h"-cobordism theorem asserted that if , "w" is compact, and "m", "n", and "w" are simply connected, then "w" is isomorphic to the cylinder (in top, pl, or diff as appropriate). this theorem proved the poincaré conjecture for .