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

TL;DR

This paper introduces a geometric approach to stable homotopy groups of spheres, providing a new proof of the Hopf Invariant One Theorem and extending the understanding of the Hopf invariant for higher dimensions.

Contribution

It develops a geometric method using Pontrjagin-Thom construction and immersion theory to prove the vanishing of the Hopf invariant in all but two specific dimensions.

Findings

New proof of Hopf Invariant One Theorem for most dimensions

Demonstrates the vanishing of the stable Hopf invariant for n>31

Uses geometric topology and immersion techniques to analyze homotopy groups

Abstract

We develop a geometric approach to stable homotopy groups of spheres in the spirit of the work of Pontrjagin and Rokhlin. A new proof of the Hopf Invariant One Theorem by J.F.Adams is obtained in all dimensions except 15 and 31. To prove that the stable Hopf invariant H: \Pi_n \to Z/2 vanishes for n>31, we apply methods of geometric topology. The Pontrjagin-Thom construction along with Hirsch's compression lemma identify every \alpha \in \Pi_n with the framed bordism class of a framed immersion of a closed n-manifold into R^{n+k}, for any given k>0. Its self-intersection M projects to an immersion f: M \to R^n which is framed by k copies of a line bundle \kappa. It is well-known that H(\alpha) = <w_1(\kappa)^{n-k}, [M]>. The self-intersection N of f is framed by k copies of a plane bundle with structure group D_4. We observe that H(\alpha) = <w_1(i^*\kappa)^{n-2k}, [\bar N]>, where i immerses the double cover \bar N of N into M. The hardest part of the proof is to show that, after modifying f in its skew-framed bordism class, the classifying map g: N \to K(D_4,1) factors through K(Z/4,1), provided that n=2^l-1, l>5 and n-2k=15. This is achieved by analyzing immersions in the regular homotopy class of f that approximate the composition of the classifying map M \to RP^{n-k}, the projection of RP^{n-k} onto the join of copies of S^1/(Z/4) (the standard sphere), and an embedding of this join in R^n. The last step is proved with the quaternions.