Homotopy properties of regular mappings into real retract rational varieties

We study homotopy properties of regular mappings from spheres into a real retract rational variety $Y$. We show that the homotopy classes which are represented by such mappings form subgroups of the homotopy groups of $Y$, and that the groups are independent of the choice of the basepoint on $Y$ as long as $Y$ is connected. We also construct regular representatives of all the Whitehead products in all the homotopy groups of $Y$.