2014/08/02 by Aravind Asok, Asok, Aravind, Brent Doran +3
Mathematics · Physics and Astronomy · #14F42 #57R60 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1408.0413
openalex publication_date 2014/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the representability of motivic spheres by smooth varieties. We show that certain explicit "split" quadric hypersurfaces have the \mathbb A1-homotopy type of motivic spheres over the integers and that the \mathbb A1-homotopy types of other motivic spheres do not contain smooth schemes as representatives. We then study some applications of these representability/non-representability results to the construction of new exotic \mathbb A1-contractible smooth schemes. Then, we study vector bundles on even dimensional "split" quadric hypersurfaces by developing an algebro-geometric variant of the classical construction of vector bundles on spheres via clutching functions.