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

2010/11/26 by Петр Михайлович Ахметьев, Petr M. Akhmet'ev, Akhmet'ev, Petr M.
Mathematics · #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AT

paper · pdf · doi:10.48550/arxiv.1011.5717

openalex publication_date 2010/11/26 · arxiv created 2012/01/26 · arxiv updated 2012/01/27 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

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^ℓ-2, an arbitrary skew-framed immersion in Euclidean n-space \Rⁿ has zero Kervaire invariant. Additionally, for ℓ ≥ 12 (i.e., for n ≥ 4094) an arbitrary skew-framed immersion in Euclidean n-space \Rⁿ has zero Kervaire invariant if this skew-framed immersion admits a compression of order 16.

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