On approximation of maps into real algebraic homogeneous spaces

2020/11/12 by Jacek Bochnak, Wojciech Kucharz, Bochnak, Jacek +1
Mathematics · #14P05 #14P25 #14P99 #Algebraic Geometry (math.AG) #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2011.06637

openalex publication_date 2020/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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. Cⁱnfinity) map f:X-->Y can be approximated by regular maps in the C^o (resp. Cⁱnfinity) 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 Cⁱnfinity map from Sⁿ into Sⁿ can be approximated by regular maps in the Cⁱnfinity 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.

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• Approximation of maps between real algebraic varieties
• Approximation and homotopy in regulous geometry
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