2. Statement of results. We will be concerned with cohomology groups having as coefficients the integers, Z, or the integers mod 2, Z2. If Mn is an w-dimensional differentiable manifold, then WíElH'ÍM", Z2) denotes the îth Stiefel-Whitney class of the tangent bundle, and WiGH^M", Z2) denotes the ith dual Stiefel-Whitney class. Our first theorem gives a necessary condition for the existence of a...