Vector bundles and regulous maps

Let be a compact nonsingular affine real algebraic variety. We prove that every pre-algebraic vector bundle on becomes algebraic after finitely many blowing ups. Using this theorem, we then prove that the Stiefel-Whitney classes of any pre-algebraic -vector bundle on are algebraic. We also derive that the Chern classes of any pre-algebraic -vector bundles and the Pontryagin classes of any pre-algebraic -vector bundle are blow- -algebraic. We also provide several results on line bundles on .