Approximation of maps into spheres by piecewise-regular maps of class $$C^k$$ C k

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Mathematische Annalen

Approximation of maps into spheres by piecewise-regular maps of class C k Marcin Bilski1 Received: 11 December 2018 © The Author(s) 2019

Abstract The aim of this paper is to prove that every continuous map from a compact subset of a real algebraic variety into a sphere can be approximated by piecewise-regular maps of class C k , where k is an arbitrary nonnegative integer. Mathematics Subject Classification 14P05 · 14P10 · 26C15

1 Introduction The problem of algebraic approximation of continuous maps into spheres has been studied for many years (cf. [1,2,10] and references therein). Since regular maps are often to rigid to approximate arbitrary continuous maps (cf. [1,3,4]), it is natural to ask whether approximation by maps from larger classes is possible. One of such classes is the class of continuous rational maps (in real algebraic geometry studied systematically for the first time in [9]). On nonsingular real algebraic varieties it coincides with the class of regulous maps (also known as continuous hereditarily rational maps cf. [5,8]). Continuous rational maps and regulous maps have attracted a lot of attention in recent years (see [5–8,10,12–14] and references therein). It has turned out, for example, that every continuous map between spheres can be approximated by continuous rational maps (see [10]). However, not every continuous map from an arbitrary compact nonsingular real algebraic variety into a sphere can be approximated by continuous rational maps (see also [10]).

Communicated by Ngaiming Mok. Research partially supported by the NCN Grant 2014/13/B/ST1/00543.

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Marcin Bilski [email protected] Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

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M. Bilski

In this paper (as in [10,11]), a real algebraic variety is, by definition, a locally ringed space isomorphic to an algebraic subset of Rm , for some m, endowed with the Zariski topology and the sheaf of real-valued regular functions. Each real algebraic variety carries also the Euclidean topology induced by the standard metric in R. Unless explicitly stated otherwise, all topological notions relating to real algebraic varieties refer to the Euclidean topology. In a recent paper [11], Kucharz introduced a class of piecewise-regular maps. We say that a continuous map f : V ⊃ X → W , where V and W are real algebraic varieties and X ⊂ V is some subset, is a piecewise-regular map if the following holds. There is a stratification S of V such that for every stratum S ∈ S the restriction of f to each connected component of S ∩ X is a regular map. (For precise definition see Section 2.) Under the assumption that X is a compact subset of a real algebraic variety, Kucharz proved that every continuous map from X into a sphere can be approximated by piecewise-regular maps (cf. [11], Theorem 1.5). Another result of [11] (see Theorem 1.6) says that if V is a compact nonsingular real algebraic variety, then each continuous map from V into a sphere is homotopic to a