Algebraic vector bundles and $p$-local A^1-homotopy theory (2008.03363v2)

Abstract: We construct manylow rank" algebraic vector bundles onsimple" smooth affine varieties of high dimension. In a related direction, we study the existence of polynomial representatives of elements in the classical (unstable) homotopy groups of spheres. Using techniques of A^1-homotopy theory, we are able to produce ``motivic" lifts of elements in classical homotopy groups of spheres; these lifts provide interesting polynomial maps of spheres and algebraic vector bundles.