  In this paper a geometric approach toward stable homotopy groups of spheres,
based on the Pontrjagin-Thom construction is proposed. From this approach a new
proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except
$15,31,63,127$ is obtained.
  It is proved that for $n>127$ in the stable homotopy group of spheres $\Pi_n$
there is no elements with Hopf invariant one. The new proof is based on
geometric topology methods. The Pontrjagin-Thom Theorem (in the form proposed
by R.Wells) about the representation of stable homotopy groups of the real
projective infinite-dimensional space (this groups is mapped onto 2-components
of stable homotopy groups of spheres by the Khan-Priddy Theorem) by cobordism
classes of immersions of codimension 1 of closed manifolds (generally speaking,
non-orientable) is considered. The Hopf Invariant is expressed as a
characteristic number of the dihedral group for the self-intersection manifold
of an immersed codimension 1 manifold that represents the given element in the
stable homotopy group. In the new proof the Geometric Control Principle (by
M.Gromov) for immersions in a given regular homotopy classes based on
Smale-Hirsch Immersion Theorem is required.
