  We study smooth maps between smooth manifolds with only fold points as their
singularities, and clarify the obstructions to the existence of such a map in a
given homotopy class for certain dimensions. The obstructions are described in
terms of characteristic classes, which arise as Postnikov invariants, and can
be interpreted as primary and secondary obstructions to the elimination of
certain singularities. We also discuss the relationship between the existence
problem of fold maps and that of vector fields of stabilized tangent bundles.
