The /z-cobordism theorem in , the generalized Poincaré conjecture in higher dimensions in  and several other results in differential topology are proved by using the following theorems of Morse theory:
(1) the elimination of critical points;
(2) the existence of nondegenerate functions for which the descending and ascending bowls have normal intersection;
(3) the alteration of function values at critical points. (For the details see below.)
We shall give short and elementary pr∞fs of these theorems together with some stronger statements than the ones given in [8-13] or .
The theorems are proved for noncompact manifolds rather than for compact manifolds since, by a trivial modification of the manifold (deleting the boundary or one point) the case of compact manifolds is included.