  A geometric approach to the stable homotopy groups of spheres is developed in
this paper, based on the Pontryagin-Thom construction. The task of this
approach is to obtain an alternative proof of the Hill-Hopkins-Ravenel theorem
[H-H-R] on Kervaire invariants in all dimensions, except, possibly, a finite
number of dimensions. In the framework of this approach, the Adams theorem on
the Hopf invariant is studied, for all dimensions with the exception of 15, 31,
63, 127. The new approach is based on the methods of geometric topology.
