Abstract
Let V, W be real algebraic varieties (that is, up to isomorphism, real algebraic sets), and X subseteq V some subset. A map from X into W is said to be regular if it can be extended to a regular map defined on some Zariski locally closed subvariety of V that contains X. Furthermore, a continuous map f :X rightarrow W is said to be piecewise-regular if there exists a stratification mathscr {S} of V such that for every stratum S in mathscr {S} the restriction of f to each connected component of X cap S is a regular map. By a stratification of V we mean a finite collection of pairwise disjoint Zariski locally closed subvarieties whose union is equal to V. Assuming that the subset X of V is compact, we prove that every continuous map from X into a Grassmann variety or a unit sphere can be approximated by piecewise-regular maps. As an application, we obtain a variant of the algebraization theorem for topological vector bundles. If the variety V is compact and nonsingular, we prove that each continuous map from V into a unit sphere is homotopic to a piecewise-regular map of class mathscr {C}^k, where k is an arbitrary nonnegative integer.
Highlights
In this paper, by a real algebraic variety we mean 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
We always assume that real algebraic varieties and their subsets are endowed with the Euclidean topology
Given a compact nonsingular real algebraic variety V and two positive integers n and k, it remains an open question whether every continuous map from V into G(Fn) or Sn can be approximated by piecewise-regular maps of class C k
Summary
By a real algebraic variety we mean 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 (such an object is called an affine real algebraic variety in [6]). Definition 1.2 Let V , W be real algebraic varieties, f : X → W a continuous map defined on some subset X ⊆ V , and S a stratification of V. Theorem 1.6 Let V be a compact nonsingular real algebraic variety, n a positive integer, and f : V → Sn a continuous map. Given a compact nonsingular real algebraic variety V and two positive integers n and k, it remains an open question whether every continuous map from V into G(Fn) or Sn can be approximated by piecewise-regular maps of class C k. If V is a compact nonsingular real algebraic variety of dimension n ≥ 1, every continuous map from V into Sn can be approximated by stratified-regular maps, cf [55, Corollary 1.3].
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