2007/10/31 by Петр Михайлович Ахметьев, Akhmet'ev, Petr M. · 1 citation
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.0710.5779
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: Πn → Z/2 vanishes for n>31, we apply methods of geometric topology. The Pontrjagin-Thom construction along with Hirsch's compression lemma identify every α∈ Πn with the framed bordism class of a framed immersion of a closed n-manifold into Rn+k, for any given k>0. Its self-intersection M projects to an immersion f: M → Rn which is framed by k copies of a line bundle κ. It is well-known that H(α) = . The self-intersection N of f is framed by k copies of a plane bundle with structure group D4. We observe that H(α) = , where i immerses the double cover 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 → K(D4,1) factors through K(Z/4,1), provided that n=2l-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 → RPn-k, the projection of RPn-k onto the join of copies of S1/(Z/4) (the standard sphere), and an embedding of this join in Rn. The last step is proved with the quaternions.