Geometric approach towards stable homotopy groups of spheres. The Hopf invariant
| dc.creator | Akhmet'ev, Petr M. | |
| dc.date | 2007-10-31 | |
| dc.date | 2009-05-07 | |
| dc.date.accessioned | 2026-07-07T13:11:58Z | |
| dc.date.available | 2026-07-07T13:11:58Z | |
| dc.description | 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: Π_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 α\in Π_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 κ. It is well-known that H(α) = <w_1(κ)^{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(α) = <w_1(i^*κ)^{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. | |
| dc.description | 99 pages, in Russian | |
| dc.identifier | https://arxiv.org/abs/0710.5779 | |
| dc.identifier | http://arxiv.org/abs/0710.5779 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/229451 | |
| dc.subject | Algebraic Topology | |
| dc.subject | Geometric Topology | |
| dc.subject | 55P42; 57R42 | |
| dc.title | Geometric approach towards stable homotopy groups of spheres. The Hopf invariant | |
| dc.type | text |