Étale Homotopy Obstructions of Arithmetic Spheres

2019/02/09 by Edo Arad, Arad, Edo, Shachar Carmeli +3
Computer Science · Mathematics · #14G05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1902.03404

openalex publication_date 2019/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a field of characteristic ≠ 2 and let X be the affine variety over K defined by the equation X: a₀x₀² + ⋯ + a_nx_n² = 1 where n≥ 0 and a_i∈ K. In this paper we compute the lowest mod 2 étale homological obstruction class to the existence of a K-rational point on X, and show that it is the cup product of the form o_n+1 = [a₀]∪⋯∪[a_n]. Our computation is an étale-homotopy analogue of the topological fact that Stiefel-Whitney classes are the homological obstructions to find a section to the unit sphere bundle of a real vector bundle.

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