Approximation and homotopy in regulous geometry

Let X,Y be nonsingular real algebraic sets. A map phi: X -> Y is said to be k-regulous, where k is a nonnegative integer, if it is of class C-k and the restriction of phi to some Zariski open dense subset of X is a regular map. Assuming that Y is uniformly rational, and k >= 1, we prove that a C-infinity map f:X -> Y can be approximated by k-regulous maps in the C(k )topology if and only if f is homotopic to a k-regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nons in-gular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking Y=S-p (the unit p-dimensional sphere), we obtain several new results on approximation of C-infinity maps from X into S-p by k-regulous maps in the C-k topology, for k >= 0