Approximation by piecewise-regular maps
Marcin Bilski, Wojciech Kucharz

TL;DR
This paper proves that smooth maps from compact subsets of real algebraic varieties into uniformly rational varieties can be approximated by piecewise-regular maps, with implications for algebraizing topological vector bundles.
Contribution
It establishes a general approximation theorem for C^l maps into uniformly rational varieties using piecewise-regular maps, extending previous approximation results.
Findings
C^l maps can be approximated by piecewise-regular maps of class C^k
Results apply to algebraization of topological vector bundles
Provides a new method for approximation in real algebraic geometry
Abstract
A real algebraic variety W of dimension m is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of R^m. Let l be any nonnegative integer. We prove that every map of class C^l from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the C^l topology by piecewise-regular maps of class C^k, where k is an arbitrary integer greater than or equal to l. Next we derive consequences regarding algebraization of topological vector bundles.
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**Approximation by piecewise-regular maps
**Marcin Bilski and Wojciech Kucharz
Abstract
A real algebraic variety of dimension is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of Let be any nonnegative integer. We prove that every map of class from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the topology by piecewise-regular maps of class where is an arbitrary integer satisfying . Next we derive consequences regarding algebraization of topological vector bundles.
**Keywords: **real algebraic variety, piecewise-regular map, approximation, uniformly rational variety, piecewise-algebraic vector bundle.
**MSC: ** 14P05, 14P99, 57R22.
1 Introduction
In this paper by a real algebraic variety we mean a locally ringed space isomorphic to an algebraic subset of for some endowed with the Zariski topology and the sheaf of real-valued regular functions (cf. [4], [19], [20]). By an algebraic embedding of a real algebraic variety into we mean a map such that is a Zariski locally closed subvariety of and is a biregular isomorphism. Each real algebraic variety is also equipped with the Euclidean topology induced by the standard metric on Unless explicitly stated otherwise, all topological notions relating to real algebraic varieties refer to the Euclidean topology.
The problem of algebraic approximation of continuous maps between real algebraic varieties has been considered by several mathematicians (see [1], [4], [10], [21] and the references therein). It is well known that such maps can be approximated by continuous semialgebraic maps in the compact-open topology. This is in general false if we want to approximate by regular maps instead of semialgebraic ones even for target varieties so simple as spheres (cf. [6], [7], [4], [5]). Therefore various intermediate classes of maps (more rigid than semialgebraic ones, but with better approximation properties than regular ones) have been investigated.
One of such classes is the class of continuous rational maps (see [18]) which on nonsingular varieties coincides with the class of regulous maps (also known as continuous hereditarily rational maps or stratified-regular maps cf. [11], [16], [22]). These maps have attracted a lot of attention in recent years (see [11], [15], [16], [19], [21], [22], [25], [29] and the references therein). It has turned out, for example, that every continuous map between spheres can be approximated by regulous ones (see [19]). However, not every continuous map from an arbitrary compact nonsingular real algebraic variety into a sphere can be approximated by regulous ones (see also [19]).
Approximation of continuous maps from any compact subsets of real algebraic varieties into spheres has been recently studied in [2]. The main result of [2] says that every such map can be approximated by quasi-regulous maps which are obtained from regulous ones by changing signs of the components on some subsets of their domains.
In the present paper we approximate maps from arbitrary compact subsets of real algebraic varieties into uniformly rational real algebraic varieties (for definition see Section 2.1 below). Uniformly rational real algebraic varieties constitute a large class containing spheres, Grassmannians (especially interesting from the point of view of the theory of vector bundles), rational nonsingular real algebraic surfaces and many others (cf. Section 2.1). Enlarging the set of target varieties requires enlarging the class of approximating maps. Namely, we work with piecewise-regular maps introduced in [20] (see Section 2.2 below) the class of which contains regulous and quasi-regulous maps mentioned above as proper subclasses (cf. [2], Corollary 1); approximating maps obtained in the present paper are neither regulous nor quasi-regulous so we do not generalize here the main results of [19] or [2]. However, we do generalize Theorems 1.3, 1.5, 1.6 of [20] and their consequences, but not Theorem 1.4 of [20].
Definition. Let be (possibly singular) Zariski locally closed subvarieties of , respectively, a compact subset of and nonnegative integers.
We say that a map is of class (or a map) if it is the restriction of some map of class (equivalently, if it is the restriction of some map where is an open neighborhood of in ). We say that a map is a * piecewise-regular map* if it is both of class and piecewise-regular.
We say that a map can be approximated by piecewise-regular maps if for every and every extension of there exists a map such that the restriction is a piecewise-regular map, and
[TABLE]
for all and with
In Section 2.3.1 below we show that the approximation is compatible with some topology, which we call the topology, on the set of all maps from to Thus a map can be approximated by piecewise-regular maps if and only if every open neighborhood of in contains a piecewise-regular map (see Claim 2.6).
If are real algebraic varieties, then the notions introduced in the definition above as well as the topology on the set can be defined by means of any algebraic embeddings of in some respectively, independently of the choice of the embeddings (cf. Section 2.3.2).
We point out that the topology coincides with the usual compact-open topology. Additionally, if is a compact submanifold of and the variety is nonsingular, then is the space of all maps in the sense of differential manifolds, equipped with the compact-open topology discussed in [13], p. 34 (which, by compactness of is the same as the Whitney topology on ).
Our main result is the following
Theorem 1.1
Let be real algebraic varieties, a compact subset of , and nonnegative integers. Assume that the variety is uniformly rational. Then every map can be approximated by piecewise-regular maps.
A natural question is whether Theorem 1.1 remains true if are algebraic subvarieties of some respectively, and is uniformly rational, but "piecewise-regular" is replaced by "piecewise-polynomial". A polynomial map from a subset of to a subset of is the restriction of a polynomial map from to We obtain piecewise-polynomial maps by substituting "polynomial" for "regular" in the definition of piecewise-regular maps. The answer to the question is negative. Namely, by the Wood theorem (see [28]), there are integers such that every polynomial map from the unit -sphere into the unit-sphere is constant. Then, clearly, every piecewise-polynomial map from into is also constant.
We point out that Theorem 1.1 does not hold if is an arbitrary (nonrational) nonsingular real algebraic variety (see [20], Example 1.9). Also, in general, the approximation is not possible if (see [20], Example 1.7).
Our second result, whose presentation is postponed until Section 4, is an application of Theorem 1.1 in the theory of vector bundles. Various aspects of the problem of algebraization of topological vector bundles on a given real algebraic variety have been studied by several mathematicians for at least the last fifty years (see the references in [4], [20], [21]). It has turned out that only for exceptional varieties every topological vector bundle on is isomorphic to an algebraic vector bundle; for example, this is the case if is the unit -sphere (see [27]), but need not be so if is merely assumed to be nonsingular and diffeomorphic to (see [3]). According to [22], for a large class of varieties , which includes all varieties homeomorphic to and which is known not to include all compact nonsingular varieties, every topological vector bundle on is isomorphic to a stratified-algebraic vector bundle (or equivalently, in view of [23], to a regulous vector bundle). By Theorem 5.10 of [20] every topological vector bundle on an arbitrary compact subset of any real algebraic variety is isomorphic to a piecewise-algebraic vector bundle. In the present article we show that the latter can be chosen of class for any nonnegative integer (see Theorem 4.3). We do not obtain an analogous sharpening of Theorem 5.11 of [20].
The organization of this paper is as follows. In Section 2, we gather preliminary material on uniformly rational varieties, piecewise-regular maps, maps and the topology. In Section 3, the proof of Theorem 1.1 is given. As already indicated, Section 4 is concerned with vector bundles.
2 Preliminaries
2.1 Uniformly rational real algebraic varieties
Definition. Let be a real algebraic variety of dimension A Zariski open subset is said to be special if it is biregularly isomorphic to a Zariski open subset of The variety is said to be uniformly rational if each point of it has a special Zariski open neighborhood.
Remark. Clearly, any uniformly rational real algebraic variety is nonsingular of pure dimension. The question whether every nonsingular rational variety is uniformly rational remains open, see [8] and [12], p. 885, for the discussion involving complex algebraic varieties.
There are several important examples of real algebraic varieties which are known to be uniformly rational:
(a) The -dimensional unit sphere
[TABLE]
Note that is biregularly isomorphic to (see [4], p. 76), hence with any point removed is isomorphic to
(b) The Grassmann variety of all vector subspaces of dimension of Note that is covered by a finite number of Zariski open sets each of which is biregularly isomorphic to (see [4], p. 71 for constructing the morphisms; analogous constructions show that for equal to the field of complex numbers or the field of quaternions, the variety is a uniformly rational real algebraic variety).
(c) Rational nonsingular real algebraic surfaces. This follows in principle by the Comessatti theorem (for which see [9], p. 257 or [17], p. 206, Theorem 30 or [26], Proposition 4.3). In particular, any rational nonsingular real algebraic surface is covered by a finite number of Zariski open subsets, each isomorphic to (cf. [24], Corollary 12).
(d) Several interesting examples can be obtained by applying the theorem saying that after blowing-ups uniformly rational varieties remain uniformly rational (see [8], [12] for the proof in the complex setting which also works over the field of real numbers).
2.2 Piecewise-regular maps
Let us recall a generalization of the notion of regular map introduced in [20].
Definition. Let be real algebraic varieties, some (nonempty) subset, and the Zariski closure of in A map is said to be regular if there is a Zariski open neighborhood of and a regular map such that
A stratification of a real algebraic variety is, by definition, a finite collection of pairwise disjoint Zariski locally closed subvarieties whose union equals
Definition. Let be real algebraic varieties, a continuous map defined on some subset and a stratification of The map is said to be piecewise -regular if for every stratum the restriction of to each connected component of is a regular map (when is nonempty). Moreover, is said to be piecewise-regular if it is piecewise -regular for some stratification of
The notion of piecewise-regular map does not depend on the ambient variety To be precise, suppose that is a Zariski locally closed subvariety of a real algebraic variety Then piecewise-regularity of does not depend on whether is viewed as a subset of or as that of (see [20], p. 1546).
The following remark is an immediate consequence of the definition.
Remark. Let be real algebraic varieties and let be any subset. Then the family of all piecewise-regular real-valued functions on constitutes a ring. Moreover, if is a piecewise-regular map and are regular maps, then the composite is a piecewise-regular map.
Let us recall the notion of nonsingular algebraic arc (cf. [15], [20]). A subset of a real algebraic variety is said to be a nonsingular algebraic arc if its Zariski closure in is an algebraic curve (that is, ), and is homeomorhpic to .
The following result coming from [20] (Theorem 2.9) will be useful in the sequel.
Theorem 2.1
*Let be real algebraic varieties, a semialgebraic subset, and a continuous semialgebraic map. Then the following conditions are equivalent:
(a) The map is piecewise-regular.
(b) For every nonsingular algebraic arc in with there exists a nonempty open subset such that is a regular map.*
Corollary 2.2
Let be any semialgebraic subset and let be a piecewise-regular function. Let be a continuous function such that Then is a piecewise regular function. In particular, the absolute value of every piecewise-regular function on is a piecewise-regular function.
*Proof. *Let be any nonsingular algebraic arc in with In view of Theorem 2.1 it is sufficient to check that there exists a nonempty open subset such that is a regular function. If then is constant so it is regular. If is not contained in the zero-set of then there is a non-empty open subset of such that on or on Since are piecewise-regular, there is a nonempty open subset of such that is regular on
2.3 maps and the topology
In this section we provide a detailed discussion of the notions of map, approximation and topology that appear in Section 1. For related constructions of topologies on map spaces we refer the reader to [1] and the references therein. Henceforth stands for a nonnegative integer.
2.3.1 Basic constructions
Let be a nonempty compact subset of and let be an arbitrary subset of . A map is said to be of class (or a map) if it is the restriction of a map Denote by the set of all maps from to Clearly,
For an arbitrary open neighborhood of the restriction map
[TABLE]
is surjective; here is the -vector space of all maps form to Define the seminorm by
[TABLE]
for where the summation is over and with We endow with the topology induced by this seminorm. For any set
[TABLE]
As varies, the sets constitute a base of open neighborhoods of
We endow with the quotient topology determined by which we call the topology. Thus a subset of is open if and only if its preimage is open in the space
Claim 2.3
The map is open.
*Proof. *It suffices to prove that for every and every the set is open in Given we can choose with Now, setting we get Indeed, if and then and hence and This proves that the set is open.
Claim 2.4
Let be a map and let be a map with Then, as varies, the sets constitute a base of open neighborhoods of in
*Proof. *This, in view of continuity of the map follows from Claim 2.3.
Claim 2.5
The topology on does not depend on the choice of the open neighborhood of
*Proof. *On the set consider the quotient topology determined by and the quotient topology determined by It suffices to show that these two topologies are identical. To this end, choose a function such that in a neighborhood of and the support of is contained in For , and we get
[TABLE]
[TABLE]
where is regarded as a map on The proof is complete in view of Claim 2.4.
We define the topology on as the induced topology by regarding as the subspace of
For the sake of clarity, we explicitly formulate the following observation which is an immediate consequence of Claims 2.4, 2.5.
Claim 2.6
*Let be a map, where is a Zariski locally closed subvariety of , and let be an integer. Then the following conditions are equivalent:
(a) The map can be approximated by piecewise-regular maps (in the sense of the Definition in Section 1 with ).
(b) Every open neighborhood of in contains a piecewise-regular map.
(c) For every there exist a extension of and a map where is an open neighborhood of in such that the restriction is a piecewise-regular map, and *
2.3.2 Constructions involving varieties
Let be real algebraic varieties, let be a compact subset of and an integer. Fix algebraic embeddings By abuse of notation, we also write for the corresponding biregular isomorphisms. Moreover, let denote the restriction of
Definition. We say that a map is of class (or a map) if the map is of class in the sense of the Definition in Section 1. We say that a map is a * piecewise-regular map* if it is both of class and piecewise-regular.
We say that a map can be approximated by piecewise-regular maps if the map can be approximated by piecewise-regular maps in the sense of the Definition in Section 1.
Note that a map is piecewise-regular if and only if the map is piecewise-regular.
The space of maps endowed with the topology is already defined. Denoting by the set of all maps from to we get a bijection
[TABLE]
and define the topology on so that becomes a homeomorphism.
Claim 2.7
The set and the topology on it do not depend on the choice of the algebraic embeddings
*Proof. *Let be some algebraic embeddings, and let denote the restriction of Since both maps are biregular isomorphisms, we get a well defined homeomorphism
[TABLE]
Clearly, this completes the proof.
Having defined the space and keeping in mind Claims 2.6, 2.7 we see that the notion of approximation by piecewise-regular maps does not depend on the choice of the algebraic embeddings and Theorem 1.1 can be restated in the following equivalent form.
Theorem 2.8
Let be real algebraic varieties, a compact subset of and nonnegative integers. Assume that the variety is uniformly rational. Then, for every map and every neighborhood of in the topology, there exists a piecewise-regular map that belongs to
In the next two claims we point out that our definition of the space is compatible with the relevant standard topological constructions.
Claim 2.9
The space with the topology coincides with the space of all continuous maps from to with the compact-open topology.
*Proof. *This follows from the Tietze extension theorem for continuous functions.
Claim 2.10
Assume that is a submanifold of and the variety is nonsingular. Then the set defined in this section coincides with the set of all maps from to in the sense of differential manifolds. Moreover, the topology on is identical with the usual compact-open topology discussed in [13], p.34.
*Proof. *We may assume that are the inclusion maps. Then are submanifolds of respectively, and using tubular neighborhoods we see that the first part of the claim holds. Clearly, in the rest of the proof it issufficient to consider the case . Let denote the space of all maps from to equipped with the compact-open topology. Let be a tubular neighborhood of in and a retraction.
The map
[TABLE]
is continuous. We will show that is open which, in view of Claim 2.3, implies that To do this, take any open subset of and observe that, by Claim 2.3, the set is an open subset of Moreover, we have Therefore, it suffices to show that is an open subset of
The map
[TABLE]
is clearly continuous. Take any Then as By continuity of there is an open neighborhood of in such that Now, for every we have so Thus, we get which proves that is an open subset of
2.3.3 A useful property of functions
In the proof of our main result we shall use the property that functions which are flat on their zero-sets remain functions after some modifications. The class of functions defined below will appear in the sequel.
Definition. Let be nonnegative integers, For any open subset of let denote the class of all functions for which
The following fact from [2] (Lemma 3) will be useful.
Lemma 2.11
Let be an open subset of Let , where are integers with and If is a continuous function such that for all then
3 Proof of Theorem 1.1
For any subset of let denote the closure of with respect to the Euclidean topology of and let denote the boundary of For any two subsets of by writing we mean that is a compact set which is contained in
The following lemma is our main tool.
Lemma 3.1
*Let be a compact set and a nonnegative integer. Then for every open neighborhood of in there are open semialgebraic neighborhoods of and a piecewise-regular function of class with and with the following properties:
(1) and are unions of connected components of nonsingular algebraic subvarieties of of pure codimension
(2) and
In particular, all partial derivatives of of order from up to vanish at every point of *
*Proof. *Let be an open neighborhood of in . Without loss of generality we may assume that is compact and semialgebraic. Note that every continuous nonnegative function can be uniformly approximated on by nonnegative polynomials. Indeed, it is sufficient to approximate , using the Weierstrass approximation theorem, by a polynomial Then approximates
Applying the Weierstrass theorem and the Sard theorem do the following. Approximate on the continuous function by a nonnegative polynomial and pick so that have the following properties: and [math] is a regular value of and and
Now define and observe that, by the previous paragraph, holds true.
Let us construct Observe that for every This is because and so, by continuity, attains all values from on Using the Sard theorem, fix such that [math] is a regular value of Put and Then as vanishes only at such that or and is a compact set as is bounded.
Note that for every Indeed, let be an arc connecting any point in with some point in By the previous paragraph, there are such that and Hence, and so, by continuity, attains all values from on Once again, using the Sard theorem, fix to be a regular value of and define
Observe that for every and we have that are in different connected components of Indeed, suppose there is an arc connecting We have and so there is a point such that Hence, and Consequently, there is a point with which means that the arc intersects a contradiction.
Define on for some nonnegative integer Then . Now define on by setting on every connected component of which has a nonempty intersection with and on the other connected components of and By Lemma 2.11, we may assume that is so large that is of class Moreover, by Corollary 2.2, in view of the fact that is piecewise-regular, we conclude that is also piecewise-regular.
Next define continuous functions
[TABLE]
Corollary 2.2 implies that these functions are piecewise-regular on . Finally, for define continuous functions
[TABLE]
Inductive application of Corollary 2.2 proves that these functions are piecewise-regular on .
By construction, we have and
[TABLE]
for and Consequently, for every we have for and for
From the previous paragraph it also follows that if at some point the function is not of class then Indeed, if then, by the previous paragraph, for every and But is of class at so is of class at for every by construction.
By the fact that is compact and again by construction, for large enough, for every Take such an and set From what we have just proved we know that is of class on possibly outside the set We check that for a large odd integer and a large integer the function
[TABLE]
is of class on
The fact that is of class at every point with can be proved as follows. Clearly, there is an open neighborhood of such that is of class on where is the zero-set of Since is a continuous semialgebraic function on , then, for sufficiently large the function is of class on To show this, it is sufficient to check that if is large enough, then every partial derivative of of order on satisfies for every
Observe that is the sum of a finite (independent of ) number of terms of the form: multiplied by a constant, where and is a continuous semialgebraic function on independent of . Then the fact that for large enough, is an immediate consequence of Proposition 2.6.4 of [4]. Hence, the function is of class and therefore is of class on
Similarly, for with there is an open neighborhood of such that the continuous semialgebraic function is of class on possibly outside the zero-set of (recall that is odd). As before, for large enough, is of class on
Moreover, it is easy to observe that satisfies for every , and all partial derivatives of up to any prescribed order vanish at every point of for large enough.
Let us define by and on , and on The fact that is of class and the previous paragraph imply that is of class and satisfies (2). Clearly, is also semialgebraic.
It remains to check that is a piecewise-regular function which follows by Theorem 2.1. Indeed, let be a nonsingular algebraic arc in First assume that Then, by construction of there is an open nonempty subset of contained in such that is a regular function. If then there is an open nonempty subset of contained in either or Then is constant, hence regular. Now the claim follows immediately.
Proof of Theorem 1.1. By Section 2.3.2, we may assume that are Zariski locally closed subvarieties of some respectively. Clearly, it is sufficient to consider the case Let denote the dimension of Fix and take some extension of belonging to The extension will also be denoted by Since has a tubular neighborhood in we may assume that for some open neighborhood of in To complete the proof it is sufficient to show that for every there is a piecewise-regular map where is an open neighborhood of in satisfying (cf. Claim 2.6 with ).
Let be the least positive integer such that the set is contained in the union of special Zariski open subsets of The proof is by induction on
If then there is a Zariski open subset of such that and there is a biregular isomorphism where is a Zariski open subset of Shrinking the open neighborhood of if necessary, we get and hence for some map Now it is sufficient to approximate using the Weierstrass approximation theorem, by a polynomial map in such a way that the following holds: the restrictions to of all partial derivatives of order up to of the components of are as close to the corresponding restrictions of the partial derivatives of the components of as we wish. Then the map equal to in some neighborhood of in is a piecewise-regular map approximating
Let and let be a family of special subsets of such that Note that, there is an open bounded semialgebraic neighborhood of in with Then the compact set has an open bounded semialgebraic neighborhood in with
By Lemma 3.1, there are open semialgebraic neighborhoods of and a piecewise-regular function of class such that and the following conditions hold:
(1) and are unions of connected components of nonsingular algebraic subvarieties of of pure codimension
(2) and
Let be an open neighborhood of Note that and
Since then, by the induction hypothesis, there is a piecewise-regular map approximating in the topology. We may assume that the approximation is close enough to ensure that Since is a special subset of there is a biregular map where is a Zariski open subset of By the inclusion we have a piecewise-regular map such that
By the definition of and the choice of we have Consequently, as above, there is a map such that
Now approximate using the Weierstrass approximation theorem, by a polynomial map in such a way that the following holds: the restrictions to of all partial derivatives of order up to of the components of are as close to the corresponding restrictions of the partial derivatives of the components of as we wish. Then the map is a piecewise-regular map approximating in the topology.
Note that and observe that and are close to each other on Therefore the formula gives a piecewise-regular map close to and to and to
Finally, let us define a semialgebraic map by:
[TABLE]
and let us show that is a piecewise-regular map and that the seminorm is small.
Clearly, all partial derivatives of order up to of the components of approximate the corresponding partial derivatives of the components of on every set of the family To show that is a map and that the seminorm is small it remains to check that for every with and every the functions
[TABLE]
can be glued along to constitute a continuous function on
To do this, observe that, by the properties of (cf. Lemma 3.1), at every (resp. at every ), the corresponding partial derivatives of the components of and of (resp. of ) are equal up to order Therefore, the corresponding partial derivatives of the components of and of (resp. of ) can be glued along (resp. ). Now the claim follows immediately.
To complete the proof it is sufficient to show that is a piecewise-regular map. Here we shall use Theorem 2.1. Let be a nonsingular algebraic arc in with If then there is an open subset of such that is contained in or in or in and then the claim follows by Theorem 2.1 and the definition of If then there is an open subset of such that is contained in or in and again the claim is a direct consequence of Theorem 2.1 and the definition of
4 piecewise-algebraic vector bundles
Piecewise-algebraic vector bundles have been introduced in [20] to which we refer the reader for details. Before stating the main result of this section we recall some terminology and facts from [4], [13] and [20].
Let denote or the field of quaternions. All -vector spaces are assumed to be left vector spaces. This plays a role if since the quaternions are noncommutative. Let be a topological -vector bundle over a topological space . By we denote the total space of and by the bundle projection. The fiber of over a point is
For any nonnegative integer , let denote the standard product -vector bundle on with total space If is a topological -vector subbundle of then where is the orthogonal complement of with respect to the standard inner product on Then the orthogonal projection onto is a topological morphism of -vector bundles.
Let be a real algebraic variety. Then can also be regarded as a real algebraic variety. By an algebraic -vector bundle on we mean an algebraic vector subbundle of for some (cf. [4], Chapters 12 and 13).
Let be a subspace of a topological space and a topological morphism of topological -vector bundles on We let denote the restriction morphism defined by for all
The following generalization of the notion of algebraic vector bundle is taken from [20].
Definition. Let be a real algebraic variety, some nonempty subset and the Zariski closure of in
An algebraic -vector bundle on is a topological -vector subbundle of for some for which there exist a Zariski open neighborhood of and an algebraic -vector subbundle of with Then is also said to be an algebraic -vector subbundle of The pair is said to be an algebraic extension of
If , are algebraic -vector bundles on then an algebraic morphism is a topological morphism such that there are algebraic extensions of respectively, and an algebraic morphism with
The following notion is also taken from [20].
Definition. Let be a real algebraic variety, some subset, and a stratification of
A piecewise -algebraic -vector bundle on is a topological -vector subbundle of for some such that for every stratum and each connected component of the restriction is an algebraic -vector subbundle of In that case, is said to be a piecewise -algebraic -vector subbundle of
If are piecewise -algebraic -vector bundles on then a piecewise -algebraic morphism is a topological morphism such that for every stratum and each connected component of the restriction is an algebraic morphism.
A piecewise-algebraic -vector bundle on is a piecewise -algebraic -vector bundle on for some stratification of
If and are piecewise-algebraic -vector bundles on then a piecewise-algebraic morphism is a piecewise -algebraic morphism for some stratification of such that both and are piecewise -algebraic -vector bundles on
In what follows denotes a nonnegative integer. The notion of a bundle of class on a smooth manifold has been discussed in [13]. We generalize it to the case where the base space is a compact subset of with the Euclidean topology induced from
Definition. A topological -vector bundle on is said to be of class if is a subbundle of for some such that there exist an open neighborhood of in and an -vector subbundle of of class satisfying In that case, is called a subbundle of of class The pair is called a extension of In particular, is an -vector bundle of class on
If are -vector bundles of class on then a morphism is a topological morphism such that there exist extensions of respectively, and a morphism with
The notions introduced in the definition above can be extended in a natural way to the setting involving real algebraic varieties. For the sake of clarity, we first recall some notation.
Let be a continuous map of topological spaces. If is a topological -vector subbundle of then the pullback is a subbundle of with total space
[TABLE]
If is a morphism, where is an -vector subbundle of then the pullback morphism is defined by where and is the canonical projection.
Definition. Let be a real algebraic variety and let be a compact subset of Fix an algebraic embedding and denote by the restriction of
A topological -vector bundle on is said to be of class if is a subbundle of , for some such that the pullback -vector bundle is a subbundle of of class
If are -vector bundles of class on then a morphism is a topological morphism such that the pullback morphism is of class
The notions just defined do not depend on the choice of the algebraic embedding This assertion readily follows since if is another algebraic embedding, then the map is a biregular isomorphism.
It is clear that if is an -vector subbundle of of class then the orthogonal projection onto is a morphism of -vector bundles of class .
Note that the category of -vector bundles of class on defined in this section is equivalent to the category of topological -vector bundles on (see [14], p. 31, Proposition 5.8 and [13], p. 92, Exercise 1).
Definition. Let be a compact subset of a real algebraic variety and a stratification of A * piecewise -algebraic -vector bundle on* is a piecewise -algebraic -vector subbundle of for some which is of class .
If are piecewise -algebraic -vector bundles on then a * piecewise -algebraic morphism* is a morphism both in the category of -vector bundles of class and the category of piecewise -algebraic -vector bundles.
A * piecewise-algebraic -vector bundle on* is a piecewise -algebraic -vector bundle on for some stratification of .
If are piecewise-algebraic -vector bundles on then a * piecewise-algebraic morphism* is a piecewise -algebraic morphism, for some stratification of such that both and are piecewise -algebraic -vector bundles on
Define to be the tautological -vector bundle on The bundle is an algebraic -vector subbundle of Let be the disjoint union of the We denote by the algebraic -vector subbundle of whose restriction to is , for
Let be a topological space and a topological -vector subbundle of Then the map defined by for all is continuous and We call the classifying map for
It follows immediately from the definition and [13] that a topological -vector subbundle of where is a compact subset of a real algebraic variety, is of class if and only if the classifying map is a map. Combining this fact with Proposition 5.6 of [20] we obtain
Proposition 4.1
*Let be a real algebraic variety, a compact subset, a stratification of and a topological -vector subbundle of for some nonnegative integer Then the following conditions are equivalent:
(a) is a piecewise -algebraic -vector subbundle of
(b) The classifying map for is a piecewise -regular map.*
The following proposition is a variant of Proposition 5.8 of [20] in the category of vector bundles of class .
Proposition 4.2
Let be a real algebraic variety, a compact subset, and a stratification of . Let be piecewise -algebraic -vector bundles on that are topologically isomorphic. Then and are also isomorphic in the category of piecewise -algebraic -vector bundles on
*Proof. *We follow the proof of Proposition 5.8 of [20]. The bundle (resp. ) is a piecewise -algebraic -vector subbundle of (resp. ) of class for some (resp. ). Since and there exists a topological morphism which transforms onto Let be the matrix representation of (cf. [20], Section 4.2). By the Weierstrass approximation theorem there is a regular map close to Then defined by
[TABLE]
is an algebraic morphism.
By the fact that is a piecewise -algebraic -vector bundle on and by Lemma 5.3 of [20], the orthogonal projection onto is a piecewise -algebraic morphism. Therefore, is a piecewise -algebraic morphism which transforms onto Consequently, the morphism determined by is bijective and piecewise -algebraic and its inverse is of class . By Lemma 5.2 of [20] we conclude that is a piecewise -algebraic isomorphism.
The following consequence of Theorem 1.1 and Propositions 4.1, 4.2 is the main result of this section.
Theorem 4.3
Let be a real algebraic variety, a compact subset, and nonnegative integers. Then each -vector bundle on of class is isomorphic to a piecewise-algebraic -vector bundle on The latter bundle is uniquely determined up to piecewise-algebraic isomorphism.
*Proof. *Let be an -vector bundle on of class Then there are a positive integer and a continuous map such that is topologically isomorphic to the pullback (cf. [14], Chapter 3, Proposition 5.8). Without loss of generality, we may assume that for some
Recall that is a uniformly rational real algebraic variety. Then, by Theorem 1.1 with we obtain that is homotopic to a piecewise-regular map hence is topologically isomorphic to the pullback (cf. [14], Chapter 3, Theorem 4.7). Since and are topologically isomorphic bundles of class then (cf. [13]) they are isomorphic. Finally, by Proposition 4.1, is a piecewise-algebraic -vector bundle on and the proof is complete by Proposition 4.2.
Acknowledgements. The second named author was partially supported by the National Science Center (Poland) under Grant number 2018/31/B/ST1/01059.
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