DOI 10.1515 / ADVGEOM.2009.024 de Gruyter 2009
Rational maps in real algebraic geometry
Wojciech Kucharz∗ (Communicated by C. Scheiderer)
Abstract. The paper deals with rational maps between real algebraic sets. We are interested in the rational maps which extend to continuous maps defined on the entire source space. In particular, we prove that every continuous map between unit spheres is homotopic to a rational map of such a type. We also establish connections with algebraic cycles and vector bundles.
Key words. Rational maps, real algebraic sets, vector bundles.
2000 Mathematics Subject Classification. 14P05, 14P25
1 Introduction
LetX ⊆RkandY ⊆R`be nonsingular irreducible algebraic sets. A mapf :X −→Y of classCr (where r = 0,1, . . . ,∞, and C0 stands for continuous) is said to be a Cr rational mapif there exist a nonempty Zariski open subsetU ofXand a regular mapϕ: U −→Y withf|U =ϕ. In that casef is completely determined byϕsinceU is dense inX. Denote byP(f)the indeterminacy locus (that is, the complement of the domain of definition) of the rational map fromX intoY represented byϕ. HenceP(f)is the smallest algebraic subset ofXfor which the restriction mapf|X\P(f) :X\P(f)−→Y is regular. The graph off|X\P(f)is a semi-algebraic subset ofX×Y, whose closure is equal to the graph off. It follows that the graph off is also a semi-algebraic subset.
In other words, everyCrrational map is a semi-algebraic map. ThusCrrational maps are natural objects in real algebraic or semi-algebraic geometry. We shall see below that they have some remarkable properties. We wish to emphasize that certainCr rational maps admit a very explicit description. Since the restrictionf|X\P(f)is a regular map, there
∗The paper was completed at the Max-Planck-Institut f¨ur Mathematik in Bonn, whose support and hospitality are gratefully acknowledged.
exist polynomialsP1, . . . , P`, QinR[T1, . . . , Tk]with P(f) ={x∈Rk |Q(x) =0},
f(x) =
P1(x)
Q(x), . . . ,P`(x) Q(x)
for allxinX\P(f),
cf. [6, Proposition 3.2.3]. Of course, in general, the value offat a point inP(f)cannot be read off from the formula given above without computing a limit. However, if it happens that f(P(f))contains at most one point, then f is given on the whole X by a finite formula. In several of our resultsCrrational maps of exactly such a type are involved.
Let us mention that we obtain a new class of maps exclusively forrfinite since any C∞ rational map is automatically regular, cf. Proposition 2.1. One readily checks that every continuous rational map fromX into Y is regular, provideddimX = 1. Some explicit examples ofCrrational maps that are not regular are given in Example 2.2. In this paper we deal with problems specific to real algebraic geometry. Any continuous rational map between complex algebraic sets is regular.
Henceforth we mainly investigate under what conditions a continuous map fromX intoY is homotopic to a Cr rational map. For maps between unit spheres the answer is definitive: no extra assumptions are necessary. More precisely, letSndenote the unit n-sphere,
Sn={(x1, . . . , xn+1)∈Rn+1|x21+· · ·+x2n+1=1}.
Theorem 1.1. Letrbe a nonnegative integer. For every pair(n, p)of positive integers, each continuous maph:Sn→Spis homotopic to aCrrational mapf :Sn→Spwith codimSnP(f)≥pandf(P(f))containing at most one point.
In the literature there are some related results concerning polynomial or regular maps between spheres. The motivation was to represent homotopy classes of maps between spheres by “simple” maps, given by finite formulas. As it turned out, this cannot always be achieved with polynomial maps. Making use of quadratic forms, Wood [25] proved that every polynomial map fromSn intoSp is a constant map, providedn ≥ 2m > p for some integerm. The question whether every continuous map from Sn into Sp is homotopic to a regular map, for all pairs(n, p), remains open and seems to be a hard problem, cf. [6, p. 367] and [22, p. 1153]. According to [6, 7, 8, 22], the answer is affirmative for some pairs(n, p)with the homotopy groupπn(Sp)nontrivial, including infinitely many such pairs for which the polynomial maps are constant.
Theorem 1.1 is proved in Section 2, which contains also results concerningCrrational maps fromXintoSp. Among the latter the following is the simplest.
Theorem 1.2. Letrbe a nonnegative integer. Assume thatXis compact anddimX =n.
Then each continuous maph:X →Snis homotopic to aCrrational mapf :X →Sn with the setP(f)finite andf(P(f))containing at most one point.
This result demonstrates thatCrrational maps are indeed more flexible than regular maps. For example, in contrast with Theorem 1.2, a continuous map fromS1×S1into S2 is homotopic to a regular map if and only if it is null homotopic, cf. [6, 8]. Some
cases, resolved by Theorem 1.2 forCrrational maps, remain open problems for regular maps. For instance, it is not known whether every continuous map fromS2×S2intoS4 is homotopic to a regular map, cf. [6, Remark 12.5.5].
In general not every continuous map fromX into Spis homotopic to aCrrational map. Some obstructions depend on the existence of homology or cohomology classes not representable by algebraic subsets. Below we make this precise.
Assume thatXis compact. Denote by
Hdalg(X,Z/2)
the subgroup of the homology groupHd(X,Z/2) generated by the homology classes represented byd-dimensional algebraic subsets ofX, cf. [4, 6, 13, 14]. As usual, set
Halgc (X,Z/2) =DX−1(Hdalg(X,Z/2)),
wherec+d = dimX andDX : Hc(X,Z/2) → Hd(X,Z/2)is the Poincar´e duality isomorphism. Given any continuous maph:X→Y, denote by
h∗:Hd(X,Z/2)→Hd(Y,Z/2) and h∗:Hc(Y,Z/2)→Hc(X,Z/2) the induced homomorphisms.
Proposition 1.3. Assume thatX andY are compact. Iff : X → Y is a continuous rational map, then
f∗(Hdalg(X,Z/2))⊆Hdalg(Y,Z/2), f∗(Halgc (Y,Z/2))⊆Halgc (X,Z/2) ford≥0andc≥0.
Proposition 1.3 is well known for regular maps (cf. [4, 6, 13, 14]), and therefore it holds for continuous maps homotopic to regular ones. There are however Cr rational maps not homotopic to regular maps, cf. Theorem 1.2 and the comment that follows it.
Denote byσp the unique generator ofHalgp (Sp,Z/2) = Hp(Sp,Z/2) ∼= Z/2. By Proposition 1.3, a necessary condition for a continuous maph : X → Sp to be homo- topic to a continuous rational map is thath∗(σp)be inHalgp (X,Z/2). In some cases this condition is also sufficient.
Corollary 1.4. Assume thatX is compact. For any continuous maph : X → S1, the following conditions are equivalent:
(a) his homotopic to a continuous rational map.
(b) his homotopic to a regular map.
(c) hcan be approximated by regular maps.
(d) h∗(σ1)is inHalg1 (X,Z/2).
Proof. It is known that Conditions (b), (c), (d) are equivalent, cf. [7, Theorem 1.4]. By Proposition 1.3, (a) implies (d), while (b) trivially implies (a). 2
Regular maps with values inS1 are extensively studied in [6, 10]. It is known that Condition (d) is an essential restriction.
Theorem 1.5. Letrbe a nonnegative integer. Assume thatX is compact anddimX = n+1. For any continuous maph:X →Sn, the following conditions are equivalent:
(a) his homotopic to aCrrational mapf :X →SnwithdimP(f)≤1andf(P(f)) containing at most one point.
(b) his homotopic to a continuous rational map.
(c) h∗(σn)is inHalgn(X,Z/2).
Other results concerning representation of homotopy classes of maps fromXintoSp byCr rational maps are contained in Sections 2 and 3, cf. Theorems 2.4, 2.5, Proposi- tion 2.8, and Corollaries 2.6, 2.7, 3.8. Among these Theorem 2.4 is crucial. All results announced above are proved in Section 2. In Section 3 we studyCrrational maps with values in Grassmannians. To this end we introduce the notion, interesting in its own right, ofCrrational structure on a vector bundle.
In order to allow for ease of exposition it will be convenient to adopt in the subse- quent sections the following convention. The termreal algebraic varietywill designate a locally ringed space isomorphic to an algebraic subset of Rn, for some n, endowed with the Zariski topology and the sheaf ofR-valued regular functions. Recall that the quasi-projective real algebraic varieties are real algebraic varieties in this sense, cf. [6, Proposition 3.2.10, Theorem 3.4.4]. Zariski closed subsets of a real algebraic variety will be often called algebraic subsets. Morphisms between real algebraic varieties will be called regular maps. Every real algebraic variety carries also the Euclidean topology, which is determined by the usual metric topology onR. Unless explicitly stated other- wise, all topological notions related to real algebraic varieties will refer to the Euclidean topology. Our standard reference on real algebraic geometry is [6].
2 Rational maps into spheres
We begin by showing thatC∞rational maps are regular maps. The argument is completely standard.
Proposition 2.1. LetX andY be nonsingular irreducible real algebraic varieties. Any C∞rational map fromX intoY is a regular map.
Proof. We may assume Y ⊆ R`, for some`, and hence reduce the proof to the case Y =R. Letf : X →Rbe aC∞rational function. There exist two regular functionsϕ andψfromXintoRwithψ−1(0)6=X andf(x) =ϕ(x)/ψ(x)for allxinX\ψ−1(0).
Thusψf =ϕonX.
Given a pointxinX, we denote byRx the local ring ofX atx, that is, the ring of germs atx(with respect to the Zariski topology) of regular functions. Since X is nonsingular and irreducible, the completionRˆxofRx, with respect to the maximal ideal ofRx, is isomorphic to the ring of formal power series innvariables overR,n= dimX.
The equalityψf =ϕimplies thatψxdividesϕxinRˆx, whereϕxandψxare the germs atxof, respectively,ϕandψ. HenceψxdividesϕxinRx, cf. [21,§1.C]. Consequently, fis a regular function in a Zariski neighborhood ofx. Sincexis an arbitrary point ofX,
it follows thatf is regular onX. 2
In view of Proposition 2.1, consideringCrrational maps we will usually assume that ris finite.
Example 2.2. Assume thatris a nonnegative integer.
(i) There exists aCrrational mapg:S2→S1withP(g)6=∅. Indeed,h:R2 →R,
h(x1, x2) = ((x4
1+x42)r+1
x21+x22 for(x1, x2)6= (0,0) 0 for(x1, x2) = (0,0)
is aCrrational function withP(h) ={(0,0)}. Defineg :S2 →S1byg =σ◦h◦π, where
π:S2→R2, π(x1, x2, x3) = (x1, x2), σ:R→S1, σ(t) =
2t
t2+1,t2−1 t2+1
.
Thengis aCrrational map withP(g) ={(0,0,1),(0,0,−1)}.
(ii) There exists aCrrational mapf :S2×S1→S1such thatfis not null homotopic andf(P(f)) =S1. Indeed, definef byf(x, z) =g(x)zfor(x, z)inS2×S1, wheregis as in (i) andS1={z∈C| |z|=1}is regarded as a multiplicative group. It follows that P(f) =P(g)×S1, and hencef(P(f)) =S1. Moreover,f is not null homotopic since for any pointxinS2, the mapS1→S1, z→f(x, z)is homotopic to the identity map.
As the first step toward our main goals, we record a simple fact concerning nonsingu- lar points of real algebraic varieties.
Lemma 2.3. Let X be a nonsingular real algebraic variety and letU be a nonempty Zariski open subset ofX. LetN be a Zariski closed subset ofU and letV be its Zariski closure inX. ThenV =N∪W, whereWis a Zariski closed subset ofXwithN∩W =∅ anddimW <dimN. In particular,N is precisely the set of nonsingular points ofV, assumingNis compact and nonsingular.
Proof. The equalitydimN = dimV is well known. SinceN =V ∩U, the setW = V\N is Zariski closed inX andW ⊆X\U. We haveV =N∪W,N∩W =∅, and dimW ≤dimV. Moreover, no irreducible component ofV is contained inW. It follows thatdimW <dimV, as required. Indeed, suppose to the contrary thatdimW = dimV and choose an irreducible componentZ of W withdimZ = dimV. Then Z is an irreducible component ofV, and we get a contradiction.
The last assertion in the lemma follows immediately. 2
Proofs of the results announced in Section 1 depend on the Pontryagin–Thom con- struction. Unless explicitly specified otherwise, all smooth (of classC∞) manifolds will be without boundary. Smooth submanifolds will be closed subsets of the ambient mani- fold. The unitp-sphereSpwill be oriented as the boundary of the unit(p+1)-disk. Recall that for any compact smooth manifoldM there is a canonical one-to-one correspondence
πp(M)−→Fp(M),
whereπp(M)is the set of homotopy classes of continuous maps fromM intoSp, and Fp(M)is the set of framed cobordism classes of framed submanifolds ofM of codi- mensionp(cf. [15, 19] for details). Given a continuous maph: M → Sp, we denote by
P T(h)
the element ofFp(M)corresponding to the homotopy class ofh.
For sake of clarity, it will be convenient to introduce some notation related to this construction. A framed submanifold ofM of codimensionpis a pair(N, F), whereN is a codimensionpsmooth submanifold ofM with trivial normal bundle ν(N), while F = (v1, . . . , vp)is ap-tuple of smooth sections ofν(N)such that(v1(x), . . . , vp(x))is a basis of the fiberν(N)xofν(N)overxfor every pointxinN. Hereν(N)is regarded as the quotient vector bundle(τ(M)|N)/τ(N), whereτ(M)(respectivelyτ(N)) is the tangent bundle toM (respectivelyN).
Given a continuous maph: M → Spand a pointyinSp, assume the existence of an open neighborhoodV ofyfor which the restriction maph|h−1(V) : h−1(V) → V is smooth and transverse toy. Choose a positively oriented basisB = (w1, . . . , wp) of the tangent spaceτ(Sp)y. Then P T(h)is represented by the framed submanifold (h−1(y), F(h, B)), whereF(h, B) = (v1, . . . , vp)and(v1(x), . . . , vp(x))is transformed onto(w1, . . . , wp)by the isomorphismν(N)x→τ(Sp)yinduced by the derivativedhx: τ(M)x→τ(Sp)yfor everyxinh−1(y).
Letϕ :M → Rp be a smooth map transverse to 0 inRpand letN be the union of some connected components ofϕ−1(0). Then we obtain a framed submanifold(N, F(ϕ)) ofM, whereF(ϕ) = (v1, . . . , vp)and(v1(x), . . . , vp(x))is transformed onto the canon- ical basis of Rp by the isomorphism ν(N)x → Rp induced by the derivative dϕx : τ(M)x→τ(Rp)0=Rpfor everyxinN.
The following is the main result of this section.
Theorem 2.4. Letrbe a nonnegative integer. For any compact nonsingular irreducible real algebraic varietyX and any continuous maph:X →Sp, the following conditions are equivalent:
(a) his homotopic to aCr rational map f : X → Sp with codimXP(f) ≥ pand f(P(f))containing at most one point.
(b) his homotopic to a continuous rational mapg:X→Spwithg(P(g))6=Sp. (c) P T(h)is represented by a framed submanifold(N, F)ofX, whereN is the set of
nonsingular points of some algebraic subset ofX.
Proof. Obviously, (a) implies (b). Suppose that (b) holds. Since g(P(g)) is a proper closed subset ofSp, it follows from Sard’s Theorem that the regular mapg|X\P(g) : X\P(g)→ Sp is transverse to some pointyinSp\g(P(g)). HenceP T(h) =P T(g) is represented by the framed submanifold (g−1(y), F(g, B)), where B is a positively oriented basis ofτ(Sp)y. Note that g−1(y) is a nonsingular Zariski closed subset of X\P(g). By Lemma 2.3,g−1(y)is precisely the set of nonsingular points of the Zariski closure ofg−1(y)inX. Consequently, (c) is satisfied with(N, F) = (g−1(y), F(g, B)).
It remains to prove that (c) implies (a). This follows immediately from Theorem 2.5 below, which contains also some additional information. 2 We introduce first some terminology. LetX be a nonsingular real algebraic variety and letSbe an algebraic subset ofX. Suppose there is a finite sequence of maps
Y =Xk πk
−−−−→ Xk−1 −−−−→ · · ·πk−1 −−−−→π2 X1 −−−−→π1 X0=X, whereπ1is the blowup ofX0at a nonsingular algebraic subset ofX0contained inSand πi+1is the blowup ofXiat a nonsingular algebraic subset ofXicontained in(π1◦ · · · ◦ πi)−1(S)fori=1, . . . , k−1. We call the composite mapπ=π1◦ · · · ◦πk :Y →X amultiblowup ofX overS. Note thatY is a nonsingular real algebraic variety and the restrictionπS :Y\π−1(S)→X\Sofπis a biregular isomorphism. In particular, ifAis an algebraic subset ofY, thenπ(A)∪Sis an algebraic subset ofX.
Theorem 2.5. LetX be a compact nonsingular irreducible real algebraic variety and let(N, F)be a framed submanifold ofX of codimensionp. Assume thatN is the set of nonsingular points of some algebraic subset ofX. Then for any nonnegative integerr, there exists aCrrational mapf :X →Spsuch that
(i) codimXP(f)≥p, (ii) f(P(f))⊆ {a}, (iii) f−1(b) =N,
(iv) f|X\P(f) :X\P(f)→Spis transverse tob, (v) (N, F(f, B))is framed cobordant to(N, F),
wherea = (0, . . . ,0,1) ∈Sp,b = (0, . . . ,0,−1)∈ SpandB is a positively oriented basis of the tangent spaceτ(Sp)b.
Proof. LetZbe the Zariski closure ofNinX. By assumption,Nis the set of nonsingular points ofZ. In particular,S :=Z\N is an algebraic subset ofX. The singularities ofZ can be resolved by a finite sequence of blowups. More precisely, by Hironaka’s resolution of singularities theorem [16], there exists a multiblowupπ :Y →X ofX overS such that the Zariski closureV of π−1(N)in Y is nonsingular. We assert V = π−1(N).
Indeed,Y\π−1(S)is a Zariski open subset ofY andπ−1(N)is a Zariski closed subset ofY\π−1(S). Consequently,dim(V\π−1(N))<dimπ−1(N)(cf. Lemma 2.3). Hence π−1(N) is dense inV in the Euclidean topology, V being nonsingular. The set N is compact, so the last observation impliesπ(V) ⊆ N, and thereforeV = π−1(N), as asserted.
In particular,Y is a compact nonsingular irreducible real algebraic variety andV is a nonsingular algebraic subset ofY. Since the restriction
πS :Y\π−1(S)→X\S
ofπis a biregular isomorphism, the normal bundle ofV inY is trivial. We endowV with a framingFV so that the framed submanifold(V, FV)ofY corresponds to(N, F)viaπS. A standard transversality argument implies the existence of a smooth mapϕ:Y →Rp such thatϕis transverse to 0 inRp,V ⊆ϕ−1(0)(in particular,V is the union of some connected components ofϕ−1(0)), and(V, F(ϕ))is framed cobordant to(V, FV). By a relative version of the Weierstrass approximation theorem [6, Lemma 12.5.5], one can find a regular mapψ : Y → Rp arbitrarily close in the C∞ topology to ϕand with V ⊆ψ−1(0). Ifψis sufficiently close toϕ, thenψis transverse to 0 inRpand(V, F(ψ)) is framed cobordant to(V, F(ϕ)). Moreover,ψ−1(0)is a nonsingular algebraic subset of Y and
ψ−1(0) =V ∪W,
whereW is a subset ofY withV ∩W = ∅. SinceV is also a nonsingular algebraic subset ofY, it follows thatW is an algebraic subset of Y, cf. [6, Proposition 3.3.17].
HenceW∪π−1(S)is an algebraic subset ofY, which implies the existence of a regular functionα:Y →Rwith
α−1(0) =W∪π−1(S).
Note thatV ∩α−1(0) = ∅. By the Łojasiewicz inequality [6, Corollary 2.6.7], there exist a neighborhoodU ofα−1(0)inY, a positive real numberc, and a positive integerk satisfying
kψ(y)k ≥cα(y)2kfor allyinU, wherek kdenotes the Euclidean norm onRp. Set
β(y) =1/α(y)2(k+`)for allyinY\α−1(0), where`is a positive integer to be determined later.
The stereographic projection
ρ:Sp\{a} →Rp is a biregular isomorphism. Defineh:Y →Spby
h(y) =
(ρ−1(β(y)ψ(y)) foryinY\α−1(0) a foryinα−1(0).
By construction, the restrictionh|Y\α−1(0)is a regular map transverse tob. Moreover, his continuous,h−1(b) =V, and(V, F(h, B))is framed cobordant to(V, F(ψ)). Thus (V, F(h, B))is framed cobordant to(V, FV).
Note thatπ(α−1(0)) =π(W)∪Sis an algebraic subset ofXwithcodimXπ(α−1(0))
≥p. Definef :X →Spby f(x) =
((h◦π−1S )(x) forxinX\π(α−1(0)) a forxinπ(α−1(0)).
Thenf|X\π(α−1(0)) is a regular map transverse tob. SinceY is compact and π : Y → X is a continuous map, it follows that f is also continuous. By construction, f−1(b) =N and(N, F(f, B))is framed cobordant to(N, F). The last assertion holds since(V, F(h, B))is framed cobordant to(V, FV), and(V, FV)corresponds to(N, F) viaπS. Clearly,P(f)⊆π(α−1(0)), and hencecodimXP(f)≥pandf(P(f))⊆ {a}.
It remains to observe thatf is of classCrif`is a sufficiently large integer. This follows from the representation of blowups in local coordinates. 2 Proof of Theorem1.1. Let(M, FM)be a framed submanifold ofSnrepresentingP T(h).
We may assume that the pointa= (0, . . . ,0,1)∈ Sn is not inM, and henceρ(M)is a smooth submanifold ofRn, whereρ: Sn\{a} →Rn is the stereographic projection.
By [2, Theorem A],ρ(M)is isotopic in Rn to the set of nonsingular points of some algebraic subsetW of Rn. Denoting by V the Zariski closure ofρ−1(W) inSn, we haveV ⊆ρ−1(W)∪ {a}. ThusM is isotopic inSn to the setN of nonsingular points ofV. Consequently, there is a framingF of N such that(N, F)is framed cobordant to(M, FM). Since(N, F)representsP T(h), the proof is complete in view of Theo-
rem 2.4. 2
Proof of Theorem1.2. It suffices to apply Theorem 2.4. 2 The other results stated in Section 1 could be proved now, but instead we derive first three consequences of Theorem 2.4. These facts are not required for the proofs of Propo- sition 1.3 and Theorem 1.5.
Corollary 2.6. LetX be a compact nonsingular irreducible real algebraic variety and letpbe a positive integer. Assume thatXis homotopically equivalent toSn,n≥1. Then for any nonnegative integerr, each continuous maph: X → Spis homotopic to aCr rational mapf :X →SpwithcodimXP(f)≥pandf(P(f))containing at most one point.
Proof. Letϕ:X→Snbe a homotopy equivalence and letg:Sn →Spbe a continuous map such thathis homotopic to g◦ϕ. By Theorem 1.1, we may assume thatg is a Crrational map withg(P(g))containing at most one point. Similarly, by Theorem 2.4, we may assume thatϕis aCrrational map withϕ(P(ϕ))containing at most one point.
Moreover, ifP(g) 6= ∅, we can chooseϕso thatϕ(P(ϕ)) ⊆ P(g). By construction, g◦ϕ:X → Spis aCrrational map withP(g◦ϕ)⊆P(ϕ)∪ϕ−1(P(g)), and hence (g◦ϕ)(P(g◦ϕ))contains at most one point. Making use of Theorem 2.4, we obtain a Crrational mapf :X →Spsatisfying the required conditions. 2 The experience following from [6, 7, 8, 9] shows that in the investigation of regular maps between real algebraic varieties, one frequently encountersF-vector bundles, where Fstands forR,CorH(the quaternions). It turns out thatF-vector bundles are useful, and in Section 3 indispensable, for the purposes of the present paper. For any algebraic variety X, algebraicF-vector subbundles of the standard trivialF-vector bundle with total space X×Fn, for somen, are called algebraicF-vector bundles onX. In other words, algebraic F-vector bundles onX correspond to finitely generated projective modules over the ring of regular functions fromX intoF, cf. [6] for other equivalent definitions. In order to
prevent any confusion, let us mention that the objects called here and in [6] algebraicF- vector bundles were called in earlier papers [4, 5, 7, 8, 9, 11] strongly algebraicF-vector bundles. AnF-vector bundle on a real algebraic variety is said to admit an algebraic structureif it is isomorphic to an algebraic F-vector bundle (cf. [5, 7, 11], where this notion is extensively studied). In this section we use exclusivelyR-vector bundles.
Ifpis one of the integers 1, 2, 4 or 8, then onSp there is a smoothR-vector bundle θp of rankp, which has a smooth section transverse to the zero section and with the zero set consisting precisely of one point (this is certainly well known, but if desired [12, Theorem 1.5], containing a more general fact, can be consulted). It readily follows that for any pointy inSp, one can find a smooth sectionvofθp, which is transverse to the zero section ofθpand satisfiesZ(v) = {y}, whereZ(v) = {z ∈Sp |v(z) = 0}is the zero set ofv.
Corollary 2.7. LetXbe a compact nonsingular irreducible real algebraic variety and let h:X →Spbe a continuous map. Assume thatpis one of the integers1,2,4, or8, and theR-vector bundleh∗θponX admits an algebraic structure. Then for any nonnegative integerr, the maphis homotopic to aCrrational mapf :X →SpwithcodimXP(f)≥ pandf(P(f))containing at most one point. In particular, the conclusion holds if every R-vector bundle of rankponXadmits an algebraic structure.
Proof. We may assume thathis a smooth map. By Sard’s Theorem,his transverse to some pointyinSp. HenceP T(h)is represented by the framed submanifold(h−1(y), F(h, B)), whereBis a positively oriented basis of the tangent spaceτ(Sp)y.
Letv:Sp→θpbe a smooth section transverse to the zero section ofθpand satisfying Z(v) = {y}. By assumption, there exist an algebraicR-vector bundle ξonX and an R-vector bundle isomorphismϕ: h∗θp →ξ. General theory of smooth vector bundles allows us to assume thatϕis a smooth isomorphism. The smooth sectionu:=ϕ◦(h∗v) : X → ξ is transverse to the zero section of ξ andZ(u) = h−1(y). One can find an algebraic sections : X → ξ close touin theC∞ topology, cf. [6, Theorem 12.3.2].
Hence,Z(s)is a nonsingular algebraic subset ofX isotopic toh−1(y). It follows that (h−1(y), F(h, B))is framed cobordant to(Z(s), F), whereF is a suitable framing of Z(s). By Theorem 2.4, there is aCrrational mapf : X → Sp satisfying the required
conditions. The proof is complete. 2
We shall now derive a consequence of Theorem 2.4 related to the cohomotopy groups.
LetXbe a compact nonsingular irreducible real algebraic variety and letpbe a positive integer. Denote by
πratp(X)
the subset ofπp(X)consisting of the homotopy classes [f], where f : X → Sp is a continuous rational map with f(P(f))containing at most one point. According to Theorem 2.4, for any nonnegative integerr, each element of πratp(X)is the homotopy class of aCrrational mapf :X →SpwithcodimXP(f)≥pandf(P(f))containing at most one point. Obviously,πprat(X)contains the subsetπalgp (X)ofπp(X)consisting of the homotopy classes of regular maps fromXintoSp(cf. [6, 8] for results concerning πalgp (X)).
IfdimX ≤2p−2, thenπp(X)is endowed with the structure of commutative group, called thepth cohomotopy group ofX, cf. [17]. It is not known whetherπalgp (X)is always a subgroup ofπp(X), cf. [6, p. 361].
Proposition 2.8. With notation as above, ifdimX≤2p−2, thenπprat(X)is a subgroup ofπp(X).
Proof. First we show thatπprat(X)is closed under addition. Let[f1]and[f2]be inπratp(X).
We may assume that fi : X → Sp is a continuous rational map with codimXP(fi)≥pandfi(P(fi))⊆ {b}for some pointbinSp,i=1,2. Set
A=P(f1)∪P(f2) and B=f1(A)∪f2(A).
SincedimP(fi)≤dimX−p≤p−2 andfiis a semi-algebraic map fori=1,2, we getdimB≤p−2, cf. [6, Theorem 2.8.8]. By Sard’s Theorem, there is a point(y1, y2)in (Sp\B)×(Sp\B)such that the restriction offitoX\Ais transverse toyifori=1,2, and the restriction of(f1, f2) :X →Sp×SptoX\Ais transverse to(y1, y2). The last condition and the assumptiondimX ≤2p−2 imply
f1−1(y1)∩f2−1(y2) =∅.
Choosing a positively oriented basisBi of the tangent space τ(Sp)yi, we obtain that P T(fi)is represented by(fi−1(yi), F(fi, Bi)). The element[f1] + [f2]ofπp(X)cor- responds to the framed cobordism class of(N, F), whereN =f1−1(y1)∪f2−1(y2)and the restriction ofF tofi−1(yi)is equal toF(fi, Bi)fori =1,2. By construction,N is a nonsingular Zariski closed subset ofX\A. According to Lemma 2.3,N is the set of nonsingular points of its Zariski closure inX, which in view of Theorem 2.4 implies that [fi] + [f2]is inπratp(X).
For any[f]inπp(X), we have−[f] = [ϕ◦f], whereϕ :Sp →Sp is an arbitrary continuous map of topological degree−1. We can chooseϕregular, and hence−[f]is in
πratp(X), provided[f]is inπprat(X). 2
Corollary 2.7 withp =1 is contained in Corollary 1.4, which is based on Proposi- tion 1.3 to be proved now. Recall that for any regular mapϕ:X →Y between compact nonsingular real algebraic varieties,
ϕ∗(Hdalg(X,Z/2))⊆Hdalg(Y,Z/2), ϕ∗(Halgc (Y,Z/2))⊆Halgc (X,Z/2) for alld≥0 andc≥0, cf. [14, Section 5] or [4].
Given a compact smooth manifold M, we denote by[M]its fundamental class in Hm(M,Z/2),m = dimM. As in Section 1,DM :H∗(M,Z/2) →H∗(M,Z/2)will denote the Poincar´e duality isomorphism.
Proof of Proposition1.3. By Hironaka’s theorem on resolution of points of indetermi- nacy [16], there is a commutative diagram
V
π
~~~~~~~~~ g
@
@@
@@
@@
X f //Y
whereπ:V →Xis a multiblowup ofX overP(f)andg:V →Y is a regular map. In particular,V is a compact nonsingular real algebraic variety andπ∗([V]) = [X].
GivenβinHdalg(X,Z/2), setα=DV(π∗(D−1X(β))). Sinceπis a regular map,αis inHdalg(V,Z/2). A simple computation involving∪and∩products yields
π∗(α) =π∗(π∗(DX−1(β))∩[V])
=DX−1(β)∩π∗([V]) =DX−1(β)∩[X] =β, and hence
f∗(β) =f∗(π∗(α)) = (f ◦π)∗(α) =g∗(α),
which in turn implies thatf∗(β)is inHdalg(Y,Z/2), the mapgbeing regular. Hence f∗(Hdalg(X,Z/2))⊆Hdalg(Y,Z/2).
For anyvinHalgc (Y,Z/2), we haveπ∗(f∗(v)) = (f ◦π)∗(v) =g∗(v), and hence DX(f∗(v)) =f∗(v)∩[X] =f∗(v)∩π∗([V])
=π∗(π∗(f∗(v))∩[V]) =π∗(g∗(v)∩[V]) =π∗(DV(g∗(v))), which in turn implies
f∗(v) =D−1X(π∗(DV(g∗(v)))).
Sinceπandgare regular maps, it follows thatf∗(v)is inHalgc (X,Z/2). Hence f∗(Halgc (Y,Z/2))⊆Halgc (X,Z/2).
The proof is complete. 2
As an immediate consequence of Proposition 1.3 we can obtain a result concerning Cr rational maps with values in real projectiveq-spacePq(R), containing Corollary 1.4 as a special case. Recall thatP1(R)is biregularly isomorphic toS1.
Corollary 2.9. For any compact nonsingular irreducible real algebraic varietyX and any continuous maph:X →Pq(R), the following conditions are equivalent:
(a) his homotopic to a continuous rational map.
(b) his homotopic to a regular map.
(c) hcan be approximated by regular maps.
(d) h∗(λ)is inHalg1 (X,Z/2), whereλis the unique generator ofH1(Pq(R),Z/2) ∼= Z/2.
Proof. It is known that Conditions (b), (c), (d) are equivalent, cf. [6, Theorems 12.4.6 and 13.3.1]. SinceH1(Pq(R),Z/2) =Halg1 (Pq(R),Z/2), by Proposition 1.3, (a) implies (d),
while (b) trivially implies (a). 2
We conclude this section by proving Theorem 1.5. First some preparation is necessary.
IfN is a compact smooth submanifold of a smooth manifoldM, we denote by[N]M
the homology class represented by N inHn(M,Z/2), n = dimN. In other words, [N]M =i∗([N]), wherei:N ,→M is the inclusion map. As usual,w1(ξ)will stand for the first Stiefel–Whitney class of anR-vector bundleξ, andw1(M) = w1(τ(M)). By a smooth curve we mean a smooth manifold of dimension 1.
LetY be a compact nonsingular real algebraic variety. An algebraic curve inY is, by definition, an algebraic subset ofY of dimension 1. We say that a smooth curveCinY can beapproximated by nonsingular algebraic curves inY if every neighborhood of the inclusion mapC ,→Y (in theC∞topology) contains a smooth embeddinge: C →Y such thate(C)is a nonsingular algebraic curve inY. A characterization of smooth curves CinY having this approximation property will be needed. It is derived for curves with trivial normal bundle since a more general result is not required in the present paper (cf.
Lemma 2.15). Results of [1] will be freely used.
Notation 2.10. In the remaining part of this sectionV will denote a compact nonsingular real algebraic variety withdimV ≥3.
Lemma 2.11. LetCbe a smooth curve inV and letAbe a nonsingular algebraic curve inV. Assume thatA∩C=∅andA∪C=∂F, whereFis a compact smooth surface- with-boundary inV with trivial normal bundle. ThenCcan be approximated by nonsin- gular algebraic curves inV.
Proof. SetdimV =n+1. There is a smooth mapf :V →Rntransverse to 0 inRnand withf−1(0) =A∪C, cf. [12, Theorem 1.12]. By the relative Weierstrass approximation theorem [6, Lemma 12.5.5], one can find a regular mapg:V →Rnclose tof in theC∞ topology and satisfyingA⊆g−1(0). Ifgis sufficiently close tof, thengis transverse to 0, and henceg−1(0)is a nonsingular algebraic curve inV. Moreover,g−1(0) =A∪B withA∩B = ∅, which implies thatB is a nonsingular algebraic curve inV, cf. [6, Proposition 3.3.17]. By construction,BapproximatesC. 2 Lemma 2.12. LetCbe a smooth curve inV. Assume that for each connected component DofC, the modulo2intersection number of[D]V andDV(w1(V))is zero. Then there is a smooth submanifoldS ofV satisfying codimVS = 1,DV(w1(V)) = [S]V, and S∩C=∅.
Proof. LetN be any smooth submanifold ofV withcodimVN =1 andDV(w1(V)) = [N]V. Note thatN∩Dconsists of an even number of points for every connected compo- nentDofC. We obtainSwith the required properties by modifyingN as in [1, p. 213]
or [10, p. 599]. 2
Lemma 2.13. LetCbe a smooth curve inV. LetSbe a smooth submanifold ofV with codimVS = 1andDV(w1(V)) = [S]V. Ifi : V\S ,→ V is the inclusion map, then every homology class in the kernel of the induced homomorphism
i∗:H1(V\S,Z/2)→H1(V,Z/2)
is of the form[B]V\S, whereBis a nonsingular algebraic curve inV,B ⊆ V\S, and B∩C=∅.
Proof. Denoting byν(S)the normal bundle ofSinV, we haveτ(S)⊕ν(S)∼=τ(V)|S.
Hencew1(S) +w1(ν(S)) =j∗(w1(V)), wherej :S ,→V is the inclusion map. Since DV(w1(V)) = [S]V, we getj∗(w1(V)) =w1(ν(S)), cf. [20, Theorem 11.3]. Therefore w1(S) =0, which means thatSis orientable. One completes the proof (the essential part
of it) by repeating the argument used in [1, Lemma 5]. 2
Lemma 2.14. LetAbe a nonsingular algebraic curve inV. Then there is a nonsingular algebraic curveBinV with[A]V = [B]V and such that each connected component ofV contains at most one connected component ofB.
Proof. It suffices to repeat the argument used in [1, Lemma 6]. 2 The last four lemmas are needed only to prove the next one, which in turn will be used in the proof of Theorem 1.5.
Lemma 2.15. LetKbe a smooth curve inV with trivial normal bundle. Assume[K]V = [A]V, whereAis a nonsingular algebraic curve inV. ThenKcan be approximated by nonsingular algebraic curves inV.
Proof. In virtue of Lemma 2.14, we may assume that each connected component ofV contains at most one connected component ofA. Applying a small isotopy toK and making use of transversality, we may assumeK∩A=∅.
We will now verify that Lemma 2.12 (withC=K∪A) is applicable.
LetLbe a connected component ofK. Since the vector bundlesτ(L)andν(L)are trivial, andτ(L)⊕ν(L)∼=τ(V)|L, we get
j∗(w1(V)) =0
wherej:L ,→V is the inclusion map. Hence a simple computation yields hw1(V)∪D−1V ([L]V),[V]i=hw1(V), D−1V ([L]V)∩[V]i=hw1(V),[L]Vi
=hw1(V), j∗([L])i=hj∗(w1(V)),[L]i=0, which means that themodulo2 intersection number of[L]V andDV(w1(V))is zero.
IfH is a connected component ofA, then either[H]V =0 or[H]V = [K0]V, where K0 is the union of some connected components ofK. Thus themodulo2 intersection number of[H]V andDV(w1(V))is zero.
Hence by Lemma 2.12, there is a smooth submanifoldS inV withcodimVS = 1, DV(w1(V)) = [S]V, andS∩(K∪A) =∅. Leti:V\S ,→V be the inclusion map and let
i∗:H1(V\S,Z/2)→H1(V,Z/2)
be the induced homomorphism. Since[K]V\S−[A]V\Sis in the kernel ofi∗, it follows from Lemma 2.13 (withC=K∪A) that[K]V\S−[A]V\S = [B]V\S, or equivalently,
[K]V\S+ [A]V\S+ [B]V\S =0
for some nonsingular algebraic curveBinV satisfyingB⊆V\SandB∩(K∪A) =∅.
The equalityDV(w1(V)) = [S]V implies thatV\Sis an orientable smooth manifold.
In particular, every smooth curve inV\S has trivial normal bundle. For any oriented smooth curveEinV\S, denote byoEits homology class inH1(V\S,Z). Endowing the curvesK, A, Bwith orientations, we get
oK+oA+oB+2v=0
for some homology classvinH1(V\S,Z). LetD0be an oriented smooth curve inV\S withoD0=vandD0∩(K∪A∪B) =∅. There is a smooth embeddinge:D0×[0,1]→ V\Ssuch that the smooth surface-with-boundaryG=e(D0×[0,1])has trivial normal bundle,e(D0× {0}) =D0, andG∩(K∪A∪B) =∅. Note that∂G=D0∪D1, where D1=e(D0× {1}). EndowingD1with an appropriate orientation, we haveoD1 =v. By Lemma 2.11, there is a nonsingular algebraic curveCinY which approximatesD0∪D1. We can chooseCsatisfyingC ⊆X\S andC∩(K∪A∪B) = ∅. OrientingCin a suitable way, we getoC =2v. Hence regardingK∪A∪B∪Cas an oriented smooth curve inV\S, we obtain
oK∪A∪B∪C=0.
It follows thatK∪A∪B∪C =∂F for some compact oriented smooth surface-with- boundaryFinV\S. We can discard the connected components ofFwith empty bound- ary. Then the orientability ofFimplies that its normal bundle is trivial. SinceA∪B∪C is a nonsingular algebraic curve inV, we complete the proof by applying Lemma 2.11. 2
The next observation is of a different nature.
Lemma 2.16. LetDbe a smooth curve inV and letBbe an algebraic(possibly singular) curve inV satisfyingD∩B=∅and[D]V = [B]V. Letρ:W →V be the blowup ofV at a pointbinV\D. IfC=ρ−1(D)andAis the strict transform ofBunderρ, then
[C]W = [A]W + [H]W,
whereH is a nonsingular algebraic curve inWsatisfyingH⊆ρ−1(b)andA∩H =∅.
Proof. SetE =ρ−1(b). The homology class[C]W −[A]W is in the kernel ofρ∗, and hence
[C]W = [A]W +j∗(v),
wherev is in H1(E,Z/2)andj : E ,→ W is the inclusion map (cf. for example [3, Lemma 2.9.3]). IfdimV = n+1, thenE is biregularly isomorphic toPn(R). Since n≥2 andA∩E is a finite set, we havej∗(v) = [H]W withH satisfying the required
conditions. 2
Proof of Theorem1.5. Clearly, (a) implies (b), and in view of Proposition 1.3, part (b) implies (c).
Assuming that (c) is satisfied, we will prove now that (a) holds. If n = 1, then (a) follows from Corollary 1.4, and therefore in what followsn ≥ 2. Without loss of generality, we may assumehto be a smooth map.
Let B be an algebraic (possibly singular) curve inX with [B]X Poincar´e dual to h∗(σn). There is a sequence of blowups
Y =Xk πk
−−−−→ Xk−1 −−−−→ · · ·πk−1 −−−−→π2 X1 π1
−−−−→ X0=X,
each blowup at one point, such that the composite mapπ=π1◦ · · · ◦πk :Y →Xis a multiblowup ofXover the setSof singular points ofB, and the strict transformAofB underπis a nonsingular algebraic curve inY.
By Sard’s Theorem,his transverse to some pointyinSn\h(B). ThusD=h−1(y) is a smooth curve inX withD∩B =∅. Since the normal bundle ofDinX is trivial and the restrictionπS : Y\π−1(S)→ X\Sofπis a biregular isomorphism, it follows that the smooth curveK=π−1(D)inY has trivial normal bundle. Moreover, in view of Lemma 2.16,
[K]Y = [A]Y + [H]Y,
whereHis a nonsingular algebraic curve inY withH∩A=∅. Hence by Lemma 2.15, there is a nonsingular algebraic curveC inY approximatingK. We may chooseC ⊆ Y\π−1(S). Thus π(C) is a smooth curve in X isotopic toD. Moreover, π(C) is a nonsingular Zariski closed subset ofX\S. By Lemma 2.3, π(C)is precisely the set of nonsingular points of its Zariski closure inX. In virtue of Theorem 2.4, (a) holds. The
proof is complete. 2
3 Rational maps into Grassmannians
LetFstand forR,CorH. When convenient we identifyFwithRd(F), whered(F) = dimRF. Denote byGn,p(F)the Grassmannian ofp-dimensional vector subspaces ofFn. As in [5, 6, 7], we shall always regardGn,p(F)as a real algebraic variety. The universal F-vector bundleγn,p(F)on Gn,p(F)is algebraic. In this section we study Crrational maps with values inGn,p(F). The main result is Theorem 3.7. In our considerationsF- vector bundles will play a crucial role. Given a topological spaceY, we denote byεkY(F) the standard trivialF-vector bundle onY with total spaceY ×Fk. Ifξis anF-vector subbundle ofεkY(F), thenξ⊥will stand for the orthogonal complement ofξwith respect to the standard scalar productFk×Fk →F; thusξ⊥is anF-vector subbundle ofεkY(F) andξ⊕ξ⊥ = εkY(F). If Y is a real algebraic variety andξ is an algebraicF-vector subbundle ofεkY(F), thenξ⊥is also an algebraicF-vector subbundle ofεkY(F).
Notation 3.1. Throughout this sectionX will denote a compact nonsingular irreducible real algebraic variety. Unless explicitly specified otherwise,rwill stand for a nonnegative integer or∞.
The following notion will play a key role.
Definition 3.2. An F-vector bundleξ on X is said to admit a Cr rational structure if there exist aCr F-vector subbundleη of εkX(F), for some k, and a nonempty Zariski open subsetU ofXsuch thatξis isomorphic toηand the restrictionη|U is an algebraic F-vector subbundle ofεkU(F).
By shrinkingU, we may require in Definition 3.2 thatη|U be algebraically trivial.
SinceUis dense inX, it follows thatξhas constant rank onX.
Of course, everyF-vector bundle onX admitting an algebraic structure admits also a Crrational structure.
Although Definition 3.2 has a somewhat technical character, we will see below that it is “natural”. For any ringA(associative with 1) denote byProj(A)the set of isomorphism classes of finitely generated projective (left)A-modules. Denote byC(X,F)the ring of all continuous functions fromX intoF. For anyF-vector bundleξonX, theC(X,F)- moduleΓ(ξ)of all continuous sections ofξis finitely generated and projective. By [23], the correspondenceξ→Γ(ξ)gives rise to a bijection
V BF(X)→Proj(C(X,F)),
whereV BF(X)is the set of isomorphism classes ofF-vector bundles onX. We regard the ringRr(X,F)of allCr rational functions fromX intoF as a subring ofC(X,F).
According to [24, Theorem 2.2], the map
Proj(Rr(X,F))→Proj(C(X,F))
determined by the correspondenceM → C(X,F)⊗M is injective. The image of this map can be described in terms ofF-vector bundles admitting aCrrational structure, cf.
Proposition 3.3. In view of Proposition 2.1,R∞(X,F)is the ring of all regular functions fromX intoF, which is usually denoted byR(X,F). By [6, Proposition 2.1.12], anF- vector bundleξonX admits an algebraic structure if and only ifΓ(ξ)is isomorphic to C(X,F)⊗Pfor some finitely generated projectiveR(X,F)-moduleP.
Proposition 3.3. For anyF-vector bundleξonX of rankp, the following conditions are equivalent:
(a) ξadmits aCrrational structure.
(b) ξis isomorphic tof∗γn,p(F), wheren ≥pandf :X → Gn,p(F)is aCrrational map.
(c) Γ(ξ)is isomorphic toC(X,F)⊗M for some finitely generated projectiveRr(X,F)- moduleM.
Proof. Assume that (a) holds. Letη andU be as in Definition 3.2. Defineg : X → Gk,p(F)by
{x} ×g(x) =the fiber ofηoverx⊆ {x} ×Fk
for allxinX. Thengis aCrmap andη =g∗γk,p(F). Thusξis isomorphic tog∗γk,p(F).
Sinceg|U : U → Gk,p(F)is a regular map (cf. [6, Proposition 3.4.7]), (b) holds with n=kandf =g.
We shall now prove that (a) implies (c). Sinceη⊕η⊥ =εkX(F), we haveP⊕P⊥ ∼= Rr(X,F)k, whereP (respectivelyP⊥) is theRr(X,F)-module of allCrsections ofη (respectivelyη⊥) that areCrrational maps fromUinto the total space ofη|U(respectively η⊥|U). It follows thatP is a finitely generated projectiveRr(X,F)-module. Moreover, theC(X,F)-modulesC(X,F)⊗PandΓ(η)are isomorphic. SinceΓ(ξ)is isomorphic to Γ(η), Condition (c) holds withM =P.
Assume now that (b) is satisfied. Note thatf∗γn,p(F)is aCrF-vector subbundle of εnX(F). LetV be a nonempty Zariski open subset ofX for which the restrictionf|V : V → Gn,p(F)is a regular map. Then(f∗γn,p(F))|V = (f|V)∗γn,p(F)is an algebraic F-vector subbundle ofεnV(F), and hence (a) holds.
It remains to prove that (c) implies (a). Assume that (c) holds. LetNbe anRr(X,F)- module such thatM⊕N is isomorphic toR(X,F)nfor somen. Without loss of gener- ality we may assume thatM andNare submodules ofR(X,F)nand
M⊕N =Rr(X,F)n.
Note that the submoduleM¯ :=C(X,F)M ofC(X,F)n is isomorphic toC(X,F)⊗M. We complete the proof by constructing anF-vector subbundleζofεnX(F)corresponding toM¯. Define a subsetEofX×Fnand a mapπ:E→Xby
E={(x, v)∈X×Fn |v= (a1(x), . . . , an(x))for some(a1, . . . , an)inM¯}, π(x, v) =x.
The fiberEx = π−1(x)of πoverxis a vector subspace of {x} ×Fn. In particular, ζ:= (E, π, X)is a family ofF-vector spaces onX. SinceM¯ is isomorphic toΓ(ξ)and rankξ=p, we havedimFEx=pfor allxinX.
Let[aij]be anm×nmatrix with entries inRr(X,F), whose rows generateM as anRr(X,F)-module. ThenEx = {x} ×Ax, whereAx is the vector subspace ofFn generated by the rows of the matrix[aij(x)]. It follows that the map
h:X →Gn,p(F), h(x) =Axfor allxinX
is well defined and of classCr. By construction,h∗γn,p(F) =ζ, which implies thatζis a CrF-vector subbundle ofεnX(F). IfWis a nonempty Zariski open subset ofXfor which the restrictionsaij|W :W →Fare regular functions for alliandj, thenh|W :W → Gn,p(F)is a regular map (cf. [6, Proposition 3.4.7]) and henceζ|W = (h|W)∗γn,p(F)is an algebraicF-vector subbundle ofεnW(F). SinceΓ(ζ) = ¯M ∼= Γ(ξ), it follows thatξis isomorphic toζ. Thusξadmits aCrrational structure and (a) holds. 2 Corollary 3.4. AnF-vector bundle onXadmits aCrrational structure if and only if it is stably equivalent to anF-vector bundle admitting aCrrational structure.
Proof. The assertion follows from Proposition 3.3 and [24, Theorem 2.2]. 2 Corollary 3.5. For anyF-vector bundleξonX, the following conditions are equivalent:
(a) ξadmits aC∞rational structure.
(b) ξadmits an algebraic structure.
Proof. SinceR∞(X,F) = R(X,F)is the ring of regular functions fromX intoF, in virtue of Proposition 3.3, (a) implies (b). It is obvious that (b) implies (a). 2 AnyF-vector bundleξ can be regarded as anR-vector bundle; to indicate that we writeξR. Ifξadmits aCrrational structure, then so doesξR. Our next result shows that Definition 3.2 imposes severe restrictions onF-vector bundles.
Theorem 3.6. Letξbe anF-vector bundle onX admitting aCrrational structure. Then thekth Stiefel–Whitney classwk(ξR)ofξRis inHalgk (X,Z/2)for allk≥0.
Proof. Letp= rankξR. By Proposition 3.3, there is an integern≥pand aCrrational mapf :X →Gn,p(R)withf∗γn,p(R)∼=ξR. This implieswk(ξR) =f∗(wk(γn,p(R))).
Since
Halgk (Gn,p(R),Z/2) =Hk(Gn,p(R),Z/2)
(cf. [6, Proposition 11.3.3]), it follows from Proposition 1.3 thatwk(ξR) is inHalgk (X,
Z/2). 2
Theorem 3.7. For anyCrmaph:X→Gn,p, the following conditions are equivalent:
(a) hcan be approximated in theCrtopology byCrrational maps.
(b) his homotopic to aCrrational map.
(c) The pullbackF-vector bundleh∗γn,p(F)onXadmits aCrrational structure.
Proof. In order to ease notation, we will writeGn,p, γn,p, εiY instead ofGn,p(F), γn,p(F), εiY(F), respectively. Obviously, (a) implies (b). If (b) holds andhis homotopic to aCr rational mapg:X→Gn,p, then theF-vector bundlesh∗γn,pandg∗γn,pare isomorphic, and hence (c) is satisfied in virtue of Proposition 3.3.
It remains to prove that (c) implies (a). Assume (c) holds andh∗γn,pis isomorphic to aCrF-vector subbundleηofεkX, whose restrictionη|Uto a nonempty Zariski open subset U of X is an algebraicF-vector subbundle ofεkU. Sinceη andh∗γn,p areCr F-vector bundles, there is aCrF-vector bundle isomorphismϕ: η →h∗γn,p. Regardingh∗γn,p as a subbundle ofεnX, we define aCrsectionw:X →Hom(η, εnX)byw(x)(e) =ϕ(e) for allxinX andein the fiberE(η)xofηoverx. Note thatw(x) :E(η)x→ {x} ×Fn is an injectiveF-linear transformation for allxinX.
Claim. There is a Cr section s : X → Hom(η, εnX), arbitrarily close towin the Cr topology, such thats|Uis an algebraic section ofHom(η, εnX)|U = Hom(η|U, εnU).
The equalityεkX =η⊕η⊥implies the existence of aCrsectionv :X →Hom(εkX, εnX)satisfyingw=ρ◦v, where
ρ: Hom(εkX, εnX)→Hom(η, εnX), ρ(ψ) =ψ|η
is aCrhomomorphism ofF-vector bundles. Sections ofHom(εkX, εnX)can be identified with maps fromXintoFkn, and hence by the Weierstrass approximation theorem, there exists an algebraic sectionu:X → Hom(εkX, εnX)close tovin theCr topology. Thus theCr sections = ρ◦u: X → Hom(η, εnX)is close tow. Sinceη|U andη⊥|U are algebraicF-vector subbundles ofεkU, the restriction
Hom(εkX, εnX)|U = Hom(εkU, εnU)→Hom(η, εnX)|U = Hom(η|U, εnU)
ofρis an algebraic homomorphism. It follows thats|U is an algebraic section. The claim is proved.
Ifsis sufficiently close tow, thens(x) :E(η)x→ {x} ×Fnis an injectiveF-linear transformation for allxinX. The mapf : X → Gn,p, defined by {x} ×f(x) = s(x)(E(η)x)for allxinX, is of classCrand its restrictionf|Uis a regular map, cf. [6, Proposition 3.4.7]. Moreover,fis close toh, providedsis close tow. Hence (a) holds. 2 Of particular interest is Theorem 3.7 for maps with values inG2,1(F). Recall that G2,1(F)is biregularly isomorphic toSd(F), whered(F) = dimR(F). Ifa= (0,1)∈F×R and ifρ : Sd(F)\{a} → Rd(F) = Fis the sterographic projection, thenαF : Sd(F) → G2,1(F), defined by
αF(x) = (
F(ρ(x),1)⊆F2 forxinSd(F)\{a}
F(1,0)⊆F2 forx=a,
is a biregular isomorphism. We will make use of theF-vector bundleγd(F):=α∗Fγ2,1(F) onSd(F).
Corollary 3.8. For anyCrmaph:X→Sd(F), the following conditions are equivalent:
(a) hcan be approximated in theCrtopology byCrrational maps.
(b) his homotopic to aCrrational map.
(c) The pullbackF-vector bundleh∗γd(F)onX admits aCrrational structure.
Proof. It suffices to apply Theorem 3.7. 2
There is a connection between Corollaries 2.7 and 3.8. The values ofd(F)are 1, 2 or 4, whenFisR,CorH, respectively. In Corollary 2.7, one can takeθp = (γp)Rforp equal to one of the integers 1, 2 or 4.
The example below shows thatF-vector bundles admitting aCrrational structure are distinct from other types ofF-vector bundles studied heretofore in literature.
Example 3.9. (i) There areF-vector bundles which do not admit aCrrational structure for anyr. Indeed, letY be a nonsingular real algebraic variety diffeomorphic toS4×S1
