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 $$**D**_{4}-framed
immersion are introduced.
It is shown that a skew-framed immersion $f:\; M(3n+q)/4\backslash looparrowright$**R**^{n}, $0\; <\; q\; <<\; n$ (in the
$(3n/4+\; \epsilon )$-range)
admits a geometric $($**Z**/2 Å
**Z**/2)-control if the characteristic class
of the skew-framing of this immersion admits a retraction of the
order $q$,
i.e., there exists a mapping $$k_{0}: M^{(3n+q)/4}
® **R**P^{3(n-q)/4}
such that this composition
$I\; \backslash circ$k_{0}: M^{(3n+q)/4}
® **R**P^{3(n-q)/4}
® **R**P^{¥}
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 $$**D**_{4}-framed
manifold admits in the regular cobordism
class (modulo odd torsion) an immersion with a $($**Z**/2 Å
**Z**/4)-structure.

Location: http://mech.math.msu.su/~fpm/eng/k07/k078/k07802h.htm

Last modified: June 13, 2008