Smooth models of motivic spheres

TL;DR

This paper investigates which motivic spheres can be represented by smooth varieties, showing explicit cases where they can and cannot, and explores applications to constructing exotic smooth schemes and vector bundles.

Contribution

It provides new results on the representability of motivic spheres by smooth schemes and develops a geometric approach to vector bundles on quadrics.

Findings

Certain split quadric hypersurfaces have the $\\mathbb A^1$-homotopy type of motivic spheres.

Some motivic spheres do not contain smooth schemes as representatives.

New constructions of exotic $\mathbb A^1$-contractible smooth schemes and vector bundles.

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.