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

Abstract

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