Smooth Models of Motivic Spheres and the Clutching Construction.

We study the representability of motivic spheres by smooth varieties. We show that certain explicit "split" quadric hypersurfaces have the A¹-homotopy type of motivic spheres over the integers and that the 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 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. [ABSTRACT FROM AUTHOR]