# Inertial h-cobordisms

It is known that an h-cobordism from M #(,r)(S('p) x S('q)), dim M (GREATERTHEQ) 5, 2 (LESSTHEQ) p (LESSTHEQ) dim M-2 is inertial provided the torsion of the h-cobordism can be represented by an r x r matrix. In this paper we generalize this result and show that, in fact, the h-cobordism can be surgered to a product cobordism. The techniques do not involve the surgery obstruction groups of C.T.C. Wall as the above class contains h-cobordisms for which the surgery obstruction element associated to the h-cobordism by means of the Rothenberg sequence is non-zero We introduce an algebraic condition on the torsion of the h-cobordism which guarantees the h-cobordism is inertial provided the dimension of the h-cobordism is 7 or 15. Here again the h-cobordism can be surgered to a product cobordism, and the techniques do not involve the surgery obstruction groups of C.T.C. Wall. These groups take into account only those properties possessed by all manifolds which differ by 4n dimensions whereas the techniques here rely on the fact that the dimension of the h-cobordism is 7 or 15 Finally, if the torsion of the h-cobordism (W;M,N) satisfies (tau) = (-1)('dim) ('W)(tau)*, for M(,0) a codimension one submanifold of M having the same fundamental group, N is obtained from M by splitting along M(,0) and regluing by an isomorphism h which is described geometrically for a class of h-cobordisms characterized by an algebraic condition on the torsion, provided the dimension of the h-cobordism is 2n + 1, n (GREATERTHEQ) 3