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

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.