FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2007, VOLUME 13, NUMBER 8, PAGES 3-15

**Geometric approach to stable homotopy groups of spheres.**

The Adams--Hopf invariants

P. M. Akhmet'ev

Abstract

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In this paper, a geometric approach to stable homotopy groups of
spheres based on the Pontryagin--Thom construction is proposed.
From this approach, a new proof of Hopf invariant one theorem of
J. F. Adams for all dimensions except $15$, $31$, $63$, and $127$ is obtained.
It is proved that for $n\; >\; 127$, in stable homotopy group of
spheres $$P_{n}, there
is no elements with Hopf invariant one.
The new proof is based on geometric topology methods.
The Pontryagin--Thom theorem (in the form proposed by R. Wells)
about the representation of stable homotopy groups of the real,
projective, infinite-dimensional space (these groups are mapped onto
$2$-components of
stable homotopy groups of spheres by the Kahn--Priddy theorem) by
cobordism classes of immersions of codimension $1$ of closed manifolds
(generally speaking, nonoriented) is considered.
The Hopf invariant is expressed as a characteristic class 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 the given regular homotopy classes based on the
Smale--Hirsh immersion theorem is required.

Location: http://mech.math.msu.su/~fpm/eng/k07/k078/k07801h.htm

Last modified: June 13, 2008