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

TL;DR

This paper introduces a geometric approach using the Pontrjagin-Thom construction to study stable homotopy groups of spheres, providing a new proof of the Hopf Invariant One Theorem and establishing the non-existence of elements with Hopf invariant one beyond dimension 127.

Contribution

It offers a novel geometric proof of the Hopf Invariant One Theorem applicable to all dimensions except 15, 31, 63, and 127, utilizing cobordism classes and immersion theory.

Findings

New proof of Hopf Invariant One Theorem for most dimensions

No elements with Hopf invariant one in stable homotopy groups beyond dimension 127

Application of geometric topology methods to homotopy group problems

Abstract

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.