Continuous rational maps into the unit 2-sphere

Investigated are continuous rational maps from a compact nonsingular real algebraic set into unit spheres. Special attention is devoted to such maps with values in the unit 2-sphere.


Introduction.
Let X and Y be nonsingular real algebraic sets. A map f : X → Y is said to be continuous rational if it is continuous and there exists a regular map ϕ : U → Y , defined on a Zariski open and dense subset U of X, with f | U = ϕ. Let P (f ) denote the smallest algebraic subset of X for which the restriction map f | X\P (f ) : X\P (f ) → Y is regular. Thus P (f ) is the indeterminacy locus of the rational map from X into Y determined by ϕ. Maps with f (P (f )) = Y will be called nice. There exist continuous rational maps which are not nice, cf. [6,Example 2.2]. Continuous rational maps form a natural intermediate class between regular and continuous semi-algebraic maps, with many specific properties, cf. [3,[5][6][7].
In [6,7], continuous rational maps with values in the unit p-sphere The main results of the present paper, Theorems 1.2 and 1.4, concern approximation of continuous maps from X into S 2 by continuous rational maps. Here approximation refers to the compact open topology.  First some preparation is necessary. Let M be a smooth (of class C ∞ ) manifold, and let N be a smooth submanifold of M of codimension p. By convention, submanifolds are assumed to be closed subsets of the ambient manifold. Let P be a smooth manifold, and let Q be a smooth submanifold of P . A continuous map f : In that case, if the normal bundle of Q in P is oriented and the normal bundle of the smooth submanifold N : The unit sphere S p is considered with a fixed orientation. Hence any point z in S p can be regarded as a smooth submanifold of S p with oriented trivial normal bundle. Let σ p denote the generator of the cohomology group H p (S p ; Z) ∼ = Z determined by the orientation of S p . In other words, σ p = [z] S p .
Assumption. In the rest of this section, the algebraic set X is assumed to be compact. Proof. Since f (P (f )) is a proper compact subset of S p , it follows from Sard's theorem that the regular map f | X\P (f ) : X\P (f ) → S p is transverse to some point z in S p \f (P (f )). Hence N := f −1 (z) is a nonsingular Zariski closed subset of X\P (f ). If V is the Zariski closure of N in X, then V \N is a Zariski closed subset of X contained in P (f ), with dim(V \N ) < dim V . Thus N = Reg(V ), the set N being compact. Since the continuous map f : X → S p is transverse to z, one has f * (σ p ) = [Reg(V )] X , provided that Reg(V ) is endowed with the orientation induced by f | X\P (f ) .
Denote by A p (X; Z) the subgroup of H p (X; Z) generated by all adapted cohomology classes. It is an open problem whether or not for a continuous rational map f : X → S p , the cohomology class f * (σ p ) is in A p (X; Z). This problem is of particular interest for p = 2 in view of the following result.

Theorem 1.2.
If h : X → S 2 is a continuous map such that the cohomology class h * (σ 2 ) is in A 2 (X; Z), then h can be approximated by continuous rational maps.
The proof is postponed until Sect. 2. Corollary 1.3. Let C 1 , . . . , C n be compact connected nonsingular real algebraic sets of dimension 1. Then any continuous map from C 1 × · · · × C n into S 2 can be approximated by continuous rational maps.
Proof. It suffices to observe that and apply Theorem 1.2.
Denote by A p (X; Z/2) the image of A p (X; Z) by the reduction modulo 2 homomorphism The reduction modulo 2 of σ p , denoted byσ p , is a generator of the cohomology group H p (S p ; Z/2) ∼ = Z/2. The proof will be given in the next section. It is not clear whether Theorems 1.2 and 1.4 can be extended to maps with values in S p for p ≥ 3.

Proofs.
For any topological space T , denote by ε k T (C) the standard trivial C-vector bundle on T with total space T × C k .
A topological C-vector bundle ξ on a compact nonsingular real algebraic set X is said to admit a rational structure if there exist a topological C-vector subbundle η of ε k X (C), for some k, and a Zariski open and dense subset U of X such that ξ is isomorphic to η and the restriction η| U is an algebraic C-vector subbundle of ε k U (C), cf. [6, Definition 3.2]. Identify S 2 with the complex projective line P 1 (C), and denote by γ 2 the C-line bundle on S 2 corresponding to the universal C-line bundle λ on P 1 (C). Explicitly, let a = (0, 1) be a point in C × R = R 3 , and let ρ : S 2 \{a} → C be the stereographic projection. Then α : S 2 → P 1 (C), α(x) = (ρ(x) : 1) for x in S 2 \{a} (1 : 0) for x = a, is a smooth diffeomorphism and γ 2 := α * λ. In particular, γ 2 is an algebraic C-vector subbundle of ε 2 S 2 (C). The first Chern class c 1 (γ 2 ) of γ 2 is a generator Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.