Geometric approach to stable homotopy groups of spheres II; Arf-Kervaire Invariants

Abstract

The Kervaire Invariant 1 Problem until recently was an open problem in algebraic topology. Hill-Hopkins-Ravenel theorem clams a negative solution of the problem for all dimensions $n=2^l-2$, $l \ge 8$. We prove the statement of Hill-Hopkins-Ravenel theorem for all dimensions $2^l-2$, $l \ge l_0$, where $l_0$ is a sufficiently great positive integer. The proof is based on the Hirsh control principle and the Compression theorem by the author. A notion internal symmetry: of Abelian (for skew-framed immersions), bi-cyclic (for $\mathbb{Z}/2^{[3]}$-framed immersions) and quaternion-cyclic structure (for $\mathbb{Z}/2^{[4]}$-framed immersions) are introduced.