Approximation of maps into spheres by regulous maps

Let $X$ be a compact real algebraic set of dimension $n$. We prove that every Euclidean continuous map from $X$ into the unit $n$-sphere can be approximated by regulous map. This strengthens and generalizes previously known results.

We begin by fixing some terminology. A real algebraic variety is a locally ringed space isomorphic to some algebraic subset of R n , for some positive integer n, endowed with the Zariski topology and the sheaf of regular functions. It is worth recalling that this class is identical with the class of quasi-projective real varieties, for more detail and information see [4]. A morphism of real algebraic varieties is called a regular map. We will be also interested in the Euclidean topology of such varieties and this is the topology we will mean, unless explicitly stated otherwise, when using topological notions. By a smooth map we understand a map of class C ∞ .
Let X be a real algebraic variety. A stratification S of X is a finite collection of pairwise disjoint Zariski locally closed subvarieties whose union is equal to X. A map f : X → Y of real algebraic varieties is said to be regulous if it is continuous and if there exists some stratification S of X such that f | S is a regular map for every S ∈ S. We denote the set of all regulous maps between X and Y by R 0 (X, Y ). We shall treat R 0 (X, Y ) as a subspace of the space C(X, Y ) of all continuous maps endowed with the compact-open topology. Note that regulous maps in the sense of our definition were called stratified-regular in [21] and the followup papers [20,22]. This definition is different but equivalent to that of [7] where the terminology was introduced, see [21, Remark 2.3] or [24].
Each regulous map is also continuous rational -i.e. f is continuous and f | X 0 is regular for some Zariski open dense subset X 0 ⊂ X. While the converse is false in general, it is true if X is nonsingular, see [11].
Let us first recall the following result contained in [15].
Theorem 1.1. Let X be a compact nonsingular real algebraic variety of dimension p ≥ 1. Then the set R 0 (X, S p ) is dense in C(X, S p ).
A related weaker result allowing for a singular X is contained in [21]: Let X be a compact real algebraic variety of dimension p ≥ 1. Then any continuous map X → S p is homotopic to a regulous map.
Our aim is to strenghten Theorem 1.2 by showing that the nonsingularity assumption of Theorem 1.1 is unnecessary.
It is well-known that analogous results do not hold if regulous maps in Theorems 1.1, 1.2, and 1.3 are replaced with regular maps. For example, a continuous map S 1 × S 1 → S 2 is homotopic to a regular map if and only if it is null-homotopic, cf. [5].

Proof of the main theorem
We shall use the concept of the algebraic cohomology classes of a real algebraic variety which we recall now. Let X be a compact nonsingular real algebraic variety. A class in H * (X; Z/2) is said to be algebraic if it is Poincaré dual to a homology class in H * (X, Z/2) represented by an algebraic subset. The set H * alg (X; Z/2) of all algebraic cohomology classes is a subring of H * (X; Z/2) and if f : X → Y is a regular map, then the induced map f * in cohomology maps H * alg (Y ; Z/2) into H * alg (X, Z/2), cf [2,4,6]. An important tool we need is [15, Lemma 2.2], which allows for controlled approximation of continuous maps into projective space by regular maps. We restate it here for convenience. Lemma 2.1. Let X be a compact nonsingular real algebraic variety and let A be a Zariski closed subvariety of X. Let f : X → P n (R) be a continuous map whose restriction f | A : A → P n (R) is a regular map. Assume that f * (H 1 (P n (R); Z/2)) ⊂ H 1 alg (X; Z/2). Then one can find a regular g : X → P n (R) arbitrarily close to f and satisfying g| A = f | A (i.e every neigborhood of f in C(X, P n (R)) contains such a map).
We are now ready to prove Theorem 1.3 Proof. Let f be any map in C(X, S p ). Treating X as a closed subset of R m for some m ∈ N, one can find a smooth map f 0 : U → S p defined on some neighborhood U of X in R m such that f 0 | X is arbitrarily close to f . Let Σ denote the singular locus of X. Then, f 0 (Σ) S p , since dim Σ < p. This allows us to approximate f 0 | Σ by regular maps using the stereographic projection and Weierstrass approximation theorem. We can therefore reduce the proof (by replacing f with suitably modified f 0 ) to the case where f is a restriction of a smooth map defined on a neighborhood of X in R m and f | Σ is regular with f (Σ) S p . Then, by Sard's theorem, there exists an s 0 ∈ S p \ f (Σ) which is a regular value of the smooth map f | X\Σ . By Hironaka's resolution of singularities theorem [9,10], there exists a finite composition of blowups π : Y → X over Σ with Y nonsingular. The restriction π : Y \ π −1 (Σ) → X \ Σ of π is then a biregular isomorphism and s 0 is a regular value of the smooth map f •π. Letting F = (f • π) −1 (s 0 ) = (f •π) −1 (s 0 ), consider the blowup of Y with center F , which we will denote σ : B(Y, F ) → Y , and the blowup τ : B(S p , s 0 ) → S p of S p over s 0 . Since dim X = p, the set F is finite as a fiber of the smooth map f •π over its regular value s 0 , hence B(Y, F ) is a real algebraic variety. Finiteness of F and the fact that f • π is also a smooth map allow us to apply [1,Lemma 2.5.9] in order to construct a smooth lifting g of f • π to a map between B(Y, F ) and B(S p , s 0 ) making the following diagram commute: Our aim for now is to find a regular map H : B(Y, F ) → B(S p , s 0 ) arbitrarily close to g in C(B(Y, F ), B(S p , s 0 )) in such a way that the mapf : X → S p making the following diagram commute will be regulous and close to f :  (1) ϕ(E) = P p−1 (R) ⊂ P p (R).