Smooth models of motivic spheres

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.

26 pages, updated version containing an algebro-geometric version of the clutching construction; comments still welcome!

References in corpus (1)

• Cancellation theorem