Geometric approach towards stable homotopy groups of spheres. The Kervaire invariant II

TL;DR

This paper introduces geometric methods to analyze stable homotopy groups of spheres, focusing on controlling self-intersections of skew-framed immersions and advancing the understanding of the Kervaire invariant problem.

Contribution

It develops new geometric control techniques for self-intersections of immersions and applies these to progress on the Kervaire invariant one problem.

Findings

Skew-framed immersions admit $ ext{Z/2} imes ext{Z/2}$-control under certain conditions.

Any $ ext{D}_4$-framed manifold can be cobordant to an immersion with $ ext{Z/2} imes ext{Z/4}$-structure for large enough dimensions.

Approach toward resolving the Kervaire Invariant One Problem is outlined.

Abstract

The notion of the geometrical $\Z/2 \oplus \Z/2$--control of self-intersection of a skew-framed immersion and the notion of the $\Z/2 \oplus \Z/4$-structure (the cyclic structure) on the self-intersection manifold of a $\D_4$-framed immersion are introduced. It is shown that a skew-framed immersion $f:M^{\frac{3n+q}{4}} \looparrowright \R^n$, $0 < q <<n$ (in the $\frac{3n}{4}+\epsilon$-range) admits a geometrical $\Z/2 \oplus \Z/2$--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 $\kappa_0: M^{\frac{3n+q}{4}} \to \RP^{\frac{3(n-q)}{4}}$, such that this composition $I \circ \kappa_0: M^{\frac{3n+q}{4}} \to \RP^{\frac{3(n-q)}{4}} \to \RP^{\infty}$ is the characteristic class of the skew-framing of $f$. Using the notion of $\Z/2 \oplus \Z/2$-control we prove that for a sufficiently great $n$, $n=2^l-2$, an arbitrary immersed $\D_4$-framed manifold admits in the regular cobordism class (modulo odd torsion) an immersion with a $\Z/2 \oplus \Z/4$-structure. In the last section we present an approach toward the Kervaire Invariant One Problem.