Polynomial or Regular Mappings with Values in Spheres

Abstract

Given a real algebraic set X, we compare the set of polynomial or regular mappings X → S ^k with the corresponding set of continuous or smooth mappings. The results concerning this comparison are very diverse. Section 1 deals with the existence of nonconstant polynomial mappings from S ⁿ into S ^k, using mainly the theory of quadratic forms. We prove Wood’s theorem, which states that if n is a power of 2 and k > n, then every polynomial mapping S ⁿ → S ^k is constant. The Hopf forms are the best known polynomial mappings S ⁿ → S ^k. In Section 2, we study the geometry of Hopf forms, which, in turn, is useful for investigating the existence of such forms. Section 3 contains results concerning the set of regular mappings with values in S¹, S ² or S ⁴. The choice of these particular spheres is related to the fact that S ¹, S ² and S ⁴ are biregularly isomorphic, respectively, to the real, complex and quaternionic projective lines. The theory of algebraic vector bundles developed in the previous chapter plays a crucial role. For example, from the fact that every topological (ℝ, ℂ or ℍ) line bundle over S ⁿ is isomorphic to an algebraic one, we deduce that ℛ(S ⁿ ,S ^k) is dense in C ^∞ (S ⁿ ,S ^k ) for k = 1,2,4. In Section 4, we study the subset of those homotopy classes of mappings X → S ^k which are represented by regular mappings. We obtain interesting results especially when fc is odd. For example, we show that every element of 2 π _n(S ^k) ⊂ π _n (S ^k ) can be represented by a regular mapping (when, in addition, n > 2k - 1). Finally, the last section contains the characterization of the n-tuples q ₁, q _n of positive integers such that every regular (resp. polynomial) mapping (math) is homotopic to a constant. For this we use some concepts from K-theory.

Throughout this chapter, we work over the field ℝ of real numbers (but almost all results of the first two sections remain valid over an arbitrary real closed field). The algebraic sets and affine real algebraic varieties considered in this chapter are thus over ℝ.