On Real Algebraic Morphisms Into Even-Dimensional Spheres

Abstract

Given affine real algebraic varieties X and Y let us denote by R( X, Y) the set of regular mappings (real algebraic morphisms) from X into Y (for definitions and notions of real algebraic geometry see [2], where the theory of real algebraic varieties is treated systematically). Our aim in this paper is to study the set M(X,S') of regular mappings from affine nonsingular real algebraic varieties X into S ', the unit sphere EZn+?1x2 = 1. Earlier we obtained several results in this direction [3], [4]; cf. also [2], Chap. 13. In particular, it was shown in [4] that given a compact connected nonsingular orientable real algebraic subset X of RP, of odd dimension k, either each smooth mapping from X to Sk is homotopic to a regular mapping, or precisely those mappings of even topological degree have this property. Now we shall study the case of regular mappings into even-dimensional spheres. Strangely enough the situation then is radically different from that mentioned above for odd-dimensional spheres. For a very large class of compact smooth manifolds of even dimension 2k, most algebraic models X of these manifolds have the property that every regular mapping from X into S2k is null homotopic (the meaning of most will be made precise later; cf. Remark 1.6, Theorem 2.1, Example 2.3). This class of manifolds contains all compact Co hypersurfaces of R2k?1*