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

2021/10/09 by Петр Михайлович Ахметьев, Akhmet'ev, Petr M.
Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2110.04542

openalex publication_date 2021/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ≥ 8. We prove the statement of Hill-Hopkins-Ravenel theorem for all dimensions 2^l-2, l ≥ l₀, where l₀ 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 ℤ/2^[3]-framed immersions) and quaternion-cyclic structure (for ℤ/2^[4]-framed immersions) are introduced.

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