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

TL;DR

This paper advances the understanding of the Kervaire Invariant 1 Problem by extending the Hill-Hopkins-Ravenel theorem to all sufficiently large dimensions of the form 2^l-2, using novel geometric and algebraic methods.

Contribution

It proves the Hill-Hopkins-Ravenel theorem for all large dimensions 2^l-2, l ≥ l_0, employing the Hirsch control principle and new notions of internal symmetry.

Findings

Confirmed the non-existence of Kervaire invariant 1 elements in large dimensions

Introduced new symmetry structures for framed immersions

Extended the theorem to all sufficiently large dimensions

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.