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Smooth models of motivic spheres and the clutching construction. (English) Zbl 1405.14050
Summary: 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.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory
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