On real algebraic morphisms into even-dimensional spheres
Let X be an affine real algebraic variety and denote by \({\mathcal R}(X,S^ n)\) the set of regular mappings from X into the sphere \(S^ n\). The authors study \({\mathcal R}(X,S^ n)\) in case \(n=2k\) is even (case n odd had been investigated by the authors in previous papers) and dim(X)\(\geq 2k\). Among other results they prove the following: if M is a compact \(C^{\infty}\quad hypersurface\) in \({\mathbb{R}}^{2k+1}\), ``most'' algebraic models X of M are such that every \(f\in {\mathcal R}(X,S^{2k})\) is null homotopic. This is a consequence of a general result: Theorem. Let M be a compact, connected \(C^{\infty}\) manifold of dimension 2k. Assume that M is the boundary of a compact \(C^{\infty}\) orientable manifold. Then there exists an affine nonsingular algebraic variety X diffeomorphic to M, such that every \(f\in {\mathcal R}(X,S^{2k})\) is null homotopic. On the other hand, for a special hypersurface M, an example of infinite family of nonisomorphic nonsingular algebraic models \(X_ i\) of M such that \({\mathcal R}(X_ i,S^{2k})\) is dense in \(C^{\infty}(X_ i,S^{2k})\) is given. Finally, cases \(n=2, 4\) are thoroughly discussed and e.g. a converse of the theorem above, in case of surfaces and of 4- dimensional orientable manifolds, is proved.
scientific article; zbMATH DE number 4103230