On approximation of maps into real algebraic homogeneous spaces

Let X be a real algebraic variety (resp. nonsingular real algebraic variety) and let Y be a homogeneous space for some linear real algebraic group. We prove that a continuous (resp. C^∞) map f:X→Y can be approximated by regular maps in the C⁰ (resp. C^∞) topology if and only if it is homotopic to a regular map. Taking Y=S^p, the unit p-dimensional sphere, we obtain solutions of several problems that have been open since the 1980's and which concern approximation of maps with values in the unit spheres. This has several consequences for approximation of maps between unit spheres. For example, we prove that for every positive integer n every C^∞ map from Sⁿ into Sⁿ can be approximated by regular maps in the C^∞ topology. Up to now such a result has only been known for five special values of n, namely, n=1,2,3,4 or 7.