On approximation of maps into real algebraic homogeneous spaces

TL;DR

The paper establishes conditions under which continuous or smooth maps from real algebraic varieties to homogeneous spaces, especially spheres, can be approximated by regular maps, solving longstanding open problems in the field.

Contribution

It proves that such maps can be approximated by regular maps if and only if they are homotopic to a regular map, extending approximation results to all spheres and smooth maps.

Findings

Any smooth map from S^n to S^n can be approximated by regular maps for all n.

The result generalizes previous special cases for n=1,2,3,4,7.

Provides solutions to open problems in approximation theory of maps into spheres.

Abstract

Let X be a compact (resp. compact and nonsingular) real algebraic variety and let Y be a homogeneous space for some linear real algebraic group. We prove that a continuous (resp. C^infinity) map f:X-->Y can be approximated by regular maps in the C^o (resp. C^infinity) 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^infinity map from S^n into S^n can be approximated by regular maps in the C^infinity 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.