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In this paper a geometric approach toward stable homotopy groups of spheres,based on the Pontrjagin-Thom construction is proposed. From this approach a newproof 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 ongeometric topology methods. The Pontrjagin-Thom Theorem (in the form proposedby R.Wells) about the representation of stable homotopy groups of the realprojective infinite-dimensional space (this groups is mapped onto 2-componentsof stable homotopy groups of spheres by the Khan-Priddy Theorem) by cobordismclasses of immersions of codimension 1 of closed manifolds (generally speaking,non-orientable) is considered. The Hopf Invariant is expressed as acharacteristic number of the dihedral group for the self-intersection manifoldof an immersed codimension 1 manifold that represents the given element in thestable homotopy group. In the new proof the Geometric Control Principle (byM.Gromov) for immersions in a given regular homotopy classes based onSmale-Hirsch Immersion Theorem is required.

Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I