In dimension 3, the Kneser conjecture, proved in [14], implies that a P.L. manifold I$‘, homotopy equivalent to P iii Q, is itself a connected sum of manifolds homotopy equivalent to P and Q. The same situation exists in dimensions greater than 5 if P and Q are simply connected [l] or even just P simply connected [15] or in odd dimensions greater than 5 if the fundamental groups of P and Q have no elements of order 2 [Ill. In fact the same situation exists in all dimensions greater than 4 if the fundamental groups of P and Q have no elements of order 2 [4, 51. This also extends to all orientable 4X+ 3 dimensional manifolds, and to all manifolds Wzkf’ for which each element g of order 2 in n,(N’) satisfies [g] n am = 0 for k odd, I for k even, wl(W) the first Stiefel-Whitney class of CVand [g] the class in H,(IV; Z,) represented byg [5, 71.