  We prove several positive results regarding representation of homotopy
classes of spheres and algebraic groups by regular mappings. Most importantly
we show that every mapping from a sphere to an orthogonal or a unitary group is
homotopic to a regular one. Furthermore we prove that algebraic homotopy
classes of spheres form a subgroup of the homotopy group, and that a similar
result holds also for cohomotopy groups of arbitrary varieties.
