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

TL;DR

This paper solves the Kervaire invariant problem for large dimensions by introducing new structures on framed immersions, proving that under certain conditions, the invariant is zero.

Contribution

It introduces abelian, bicyclic, and quaternionic-cyclic structures on framed immersions and proves the Kervaire invariant vanishes in high dimensions under specific conditions.

Findings

Kervaire invariant is zero for large n=2^l-2

Skew-framed immersions with compression of order 16 have zero invariant

Results apply for n ≥ 4094 when l ≥ 12

Abstract

A solution to the Kervaire invariant problem is presented. We introduce the concepts of abelian structure on skew-framed immersions, bicyclic structure on $\Z/2^{[3]}$--framed immersions, and quaternionic-cyclic structure on $\Z/2^{[4]}$--framed immersions. Using these concepts, we prove that for sufficiently large $n$, $n=2^{\ell}-2$, an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant. Additionally, for $\ell \ge 12$ (i.e., for $n \ge 4094$) an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant if this skew-framed immersion admits a compression of order 16.