Approximation and homotopy in regulous geometry

TL;DR

This paper studies the approximation of smooth maps between real algebraic sets by k-regulous maps, establishing conditions under which such approximations are possible, especially when the target is uniformly rational.

Contribution

It proves that smooth maps can be approximated by k-regulous maps if and only if they are homotopic, expanding approximation results to a broad class of algebraic varieties.

Findings

Approximation of smooth maps by k-regulous maps is characterized by homotopy.

Includes new approximation results for maps into spheres.

Shows stability of uniformly rational varieties under blow-ups.

Abstract

Let X, Y be nonsingular real algebraic sets. A map fi: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 fi to some Zariski open dense subset of X is a regular map. Assuming that Y is uniformly rational, and k>0, we prove that a C^inf 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 real rational 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^inf maps from X into S^p by k-regulous maps in the C^k topology, for k nonnegative.