Smooth Models of Motivic Spheres and the Clutching Construction. (7th June 2016)

• Record Type: Journal Article
• Authors: Asok, Aravind; Doran, Brent; Fasel, Jean
• Is Part Of: International mathematics research notices. Volume 2017:Number 6(2017)
• Page Start: 1890
• Page End: 1925
• Publication Date: 2016-06-07
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnw065
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We study the representability of motivic spheres by smooth varieties. We show that certain explicit "split" quadric hypersurfaces have the ${\mathbb A}^1$ -homotopy type of motivic spheres over the integers and that the ${\mathbb A}^1$ -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 ${\mathbb A}^1$ -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.