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.