Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I
| dc.creator | Akhmet'ev, Petr M. | |
| dc.date | 2008-01-09 | |
| dc.date.accessioned | 2026-07-07T08:53:29Z | |
| dc.date.available | 2026-07-07T08:53:29Z | |
| dc.description | 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 $Π_n$ 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. | |
| dc.identifier | https://arxiv.org/abs/0801.1412 | |
| dc.identifier | http://arxiv.org/abs/0801.1412 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/145641 | |
| dc.subject | Geometric Topology | |
| dc.subject | Algebraic Topology | |
| dc.title | Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I | |
| dc.type | text |