FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2007, VOLUME 13, NUMBER 8, PAGES 17-41

Geometric approach to stable homotopy groups of spheres.
Kervaire invariants. II

P. M. Akhmet'ev

Abstract

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We present an approach to the Kervaire-invariant-one problem. The notion of the geometric $($Z/2 Å Z/2)-control of self-intersection of a skew-framed immersion and the notion of the $($Z/2 Å Z/4)-structure on the self-intersection manifold of a $$D₄-framed immersion are introduced. It is shown that a skew-framed immersion $f:\; M(3n+q)/4\backslash looparrowright$Rⁿ, (in the $(3n/4+\; \epsilon )$-range) admits a geometric $($Z/2 Å Z/2)-control if the characteristic class of the skew-framing of this immersion admits a retraction of the order $q$, i.e., there exists a mapping $$k₀: M^(3n+q)/4 ® RP^3(n-q)/4 such that this composition $I\; \backslash circ$k₀: M^(3n+q)/4 ® RP^3(n-q)/4 ® RP^¥ is the characteristic class of the skew-framing of $f$. Using the notion of $($Z/2 Å Z/2)-control, we prove that for a sufficiently large $n$, $n=2l$- 2, an arbitrary immersed $$D₄-framed manifold admits in the regular cobordism class (modulo odd torsion) an immersion with a $($Z/2 Å Z/4)-structure.

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Last modified: June 13, 2008