Algebraic approximation of mappings into spheres (Q1094487)

This paper is concerned with approximation of differentiable mappings into spheres by polynomial or rational regular functions (in the \(C^{\infty}\) topology). Let \(X\subseteq R^ n\) and \(Y\subseteq R^ p\) be real algebraic sets. Denote by \({\mathcal R}(X,Y)\) the set of rational maps \((f_ 1/g_ 1,...,f_ p/g_ p)\) between X and Y such that \(f_ i^{-1}(0)\cap X=\emptyset\) for \(i=1,...,p\). \({\mathcal E}(X,Y)\) is the set of smooth maps with \(C^{\infty}\) topology. Theorem 1: For each \(n\in {\mathbb{N}}\), and for \(k=1,2,4\), \({\mathcal R}(S^ n,S^ k)\) is dense in \({\mathcal E}(S^ n,S^ k).\) Theorem 2: Let X be a compact non singular real algebraic set and let f be a smooth map from X to \(S^ k\). If \(k=1,2,4\) the following conditions are equivalent: (1) f can be approximated in \({\mathcal E}(X,S^ k)\) by elements of \({\mathcal R}(X,S^ k)\); (2) f is homotopic to an element of \({\mathcal R}(X,S^ k).\) Theorem 3: (as corollary of a more general theorem on maps between X and \(S^ 1\) and relations between the homology and algebraic homology groups of X): Let X be a non singular compact real algebraic curve; then \({\mathcal R}(X,S^ 1)\) is dense in \({\mathcal E}(X,S^ 1).\) Theorem 4: Let X be a non orientable compact connected real algebraic surface. Then \({\mathcal R}(X,S^ 2)\) is dense in \({\mathcal E}(X,S^ 2)\) if one of the following conditions holds: (1) there exist a non singular algebraic curve \(C\subset X\) such that {\#}\({}_ 2(C,C;X)=2\) \((=\) the modulo 2 self-intersection number of C in X); (2) \(H_ 1^{alg}(X,Z_ 2)=H_ 1(X,Z_ 2);\) (3) the genus of X (as a smooth surface) is odd. The authors give also many examples.