
TL;DR
This paper introduces the concept of piecewise-regular maps between real algebraic varieties, demonstrating that continuous maps can be approximated or homotoped by such maps under certain conditions, with applications to vector bundle algebraization.
Contribution
It establishes the approximation of continuous maps by piecewise-regular maps on algebraic varieties, extending the algebraization of topological vector bundles.
Findings
Continuous maps into Grassmannians or spheres can be approximated by piecewise-regular maps.
Every continuous map into a sphere on a compact nonsingular variety is homotopic to a C^k piecewise-regular map.
Application to algebraization theorem for topological vector bundles.
Abstract
Let V, W be real algebraic varieties (that is, up to isomorphism, real algebraic sets), and let X be a subset of V. A map f from X into W is said to be regular if it can be extended to a regular map defined on some Zariski locally closed subvariety of V that contains X. Furthermore, such a map is said to be piecewise-regular if there exists a stratification of V such that the restriction of f to the intersection of X with each stratum is a regular map. By a stratification of V we mean a finite collection of pairwise disjoint Zariski locally closed subvarieties whose union is equal to V. Assuming that the subset X is compact, we prove that every continuous map from X into a Grassmann variety or a unit sphere can be approximated by piecewise-regular maps. As an application, we obtain a variant of the algebraization theorem for topological vector bundles. If the variety V is compact andâŠ
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory · Magnolia and Illicium research
\abslabeldelim
.
Piecewise-regular maps
Wojciech Kucharz The author was partially supported by National Science Centre (Poland) under grant number 2014/15/B/ST1/00046.
Abstract
Let , be real algebraic varieties (that is, up to isomorphism, real algebraic sets), and some subset. A map from into is said to be regular if it can be extended to a regular map defined on some Zariski locally closed subvariety of that contains . Furthermore, a continuous map is said to be piecewise-regular if there exists a stratification of such that for every stratum the restriction of to each connected component of is a regular map. By a stratification of we mean a finite collection of pairwise disjoint Zariski locally closed subvarieties whose union is equal to . Assuming that the subset is compact, we prove that every continuous map from into a Grassmann variety or a unit sphere can be approximated by piecewise-regular maps. As an application, we obtain a variant of the algebraization theorem for topological vector bundles. If the variety is compact and nonsingular, we prove that each continuous map from into a unit sphere is homotopic to a piecewise-regular map of class , where is an arbitrary nonnegative integer.
{@bstr@ctlist}\abstractnamefont
Key words@bslabeldelim\abstracttextfont Real algebraic variety, regular map, piecewise-regular map, approximation, vector bundle, simplicial complex.
{@bstr@ctlist}\abstractnamefont
Mathematics subject classification (2010)@bslabeldelim\abstracttextfont 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 (such an object is called an affine real algebraic variety in [7]). The class of real algebraic varieties is identical with the class of quasi-projective real algebraic varieties, cf. [7, Proposition 3.2.10, Theorem 3.4.4]. Morphisms of real algebraic varieties are called regular maps. Each real algebraic variety carries also the Euclidean topology, induced by the standard metric on . Unless explicitly stated otherwise, we always refer to the Euclidean topology.
Topological properties of regular maps and their applications to algebraization of topological vector bundles were investigated in numerous papers [3â15,17â26,28,32â36,38â41,45â47,49,51,54,62,66,69,70,74â78,80,81]. In general, regular maps are too rigid to reflect adequately topological phenomena. It is therefore desirable to introduce maps which have many good features of regular maps but are more flexible.
We first generalize the definition of regular map.
Definition 1.1**.**
Let , be real algebraic varieties, some subset, and the Zariski closure of in .
A map is said to be regular if there exist a Zariski open neighborhood of and a regular map such that .
The next step requires the concept of stratification. By a stratification of a real algebraic variety we mean a finite collection of pairwise disjoint Zariski locally closed subvarieties (some possibly empty) whose union is equal to .
Definition 1.2**.**
Let , be real algebraic varieties, a continuous map defined on some subset , and a stratification of .
The map is said to be -regular if for every stratum the restriction of to is a regular map. Also, is said to be piecewise -regular if for every stratum the restriction of to each connected component of is a regular map.
Moreover, is said to be stratified-regular (resp. piecewise-regular) if it is -regular (resp. piecewise -regular) for some stratification of .
Essentially, these notions do not depend on the ambient variety . More precisely, suppose that is a Zariski locally closed subvariety of a real algebraic variety . The map is -regular (resp. piecewise -regular) if and only if it is -regular (resp. piecewise -regular), where is the stratification of defined by with the Zariski closure of in . Conversely, given a stratification of , the map is -regular (resp. piecewise -regular) if and only if it is -regular (resp. piecewise -regular), where is the stratification of defined by . Thus, in the definition of stratified-regular (resp. piecewise-regular) it does not matter whether is regarded as a subset of or as a subset of .
Evidently, each stratified-regular map is piecewise-regular, whereas the converse is not always true. General properties of these two classes of maps and relationships between them are discussed in Section 2. Stratified-regular maps and functions are thoroughly investigated, in a more restrictive framework, in [5, 29, 30, 42, 43, 48, 50, 52, 53, 56, 58, 59, 60, 64, 72, 73, 82], where they sometimes appear under different names (cf. Remark 2.5). If is a semialgebraic set, then each piecewise-regular map defined on is semialgebraic. Theorem 2.9 provides a nontrivial characterization of piecewise-regular maps among semialgebraic ones.
In this section we concentrate on topological properties of piecewise-regular maps. With notation as in Definition 1.2, we let denote the space of all continuous maps from into , endowed with the compact-open topology. We say that the map can be approximated by piecewise -regular maps if every neighborhood of in contains a piecewise -regular map. Approximation of by maps of other types (regular, -regular, stratified-regular, piecewise-regular, etc.) is defined in the analogous way.
We pay special attention to maps with values in Grassmannians. We let stand for , or (the quaternions). When convenient, will be identified with , where . We will consider only left -vector spaces, which plays a role if since the quaternions are noncommutative. For any integers and , with , we denote by the Grassmann space of -dimensional -vector subspaces of . As in [7, Sections 3.4 and 13.3], will be regarded as a real algebraic variety. The disjoint union
[TABLE]
is also a real algebraic variety.
Theorem 1.3**.**
Let be a real algebraic variety and let be a compact subset. Then, for each positive integer , every continuous map from into can be approximated by piecewise-regular maps.
Under an additional assumption on , we also have a stronger result.
Theorem 1.4**.**
Let be a real algebraic variety and let be a compact locally contractible subset. Then there exists a stratification of such that, for each positive integer , every continuous map from into can be approximated by piecewise -regular maps.
Virtually all topological spaces one encounters in real algebraic geometry are locally contractible; for example, any semialgebraic set is locally contractible, cf. [7, Theorem 9.3.6].
The proofs of Theorems 1.3 and 1.4, given in Section 4, are based on some fairly explicit constructions.
It is well-known that maps with values in encode information on algebraic and topological -vector bundles, cf. [7, 37]. This is also the case for stratified-algebraic -vector bundles introduced in [59] and further investigated in [55, 57, 61, 64]. Theorems 1.3 and 1.4 have a bearing on -vector bundles as well, which is elaborated upon in Section 5. The main results of Section 5 are Theorems 5.10 and 5.11.
We also have an approximation theorem for maps with values in the unit -sphere
[TABLE]
Theorem 1.5**.**
Let be a real algebraic variety and let be a compact subset. Then, for each positive integer , every continuous map from into can be approximated by piecewise-regular maps.
Theorem 1.5, which is proved in Section 6, implies that each continuous map from into is homotopic to a piecewise-regular map. However, the following homotopy result requires a different proof.
Theorem 1.6**.**
Let be a compact nonsingular real algebraic variety, a positive integer, and a continuous map. Then there exists a stratification of such that, for each nonnegative integer , the map is homotopic to a piecewise -regular map of class .
We prove Theorem 1.6 in Section 7. It turns out that a suitable stratification of is quite simple and consists of at most 3 strata.
Theorems 1.3, 1.5 and 1.6 are optimal in the sense explained in the following example.
Example 1.7**.**
Let be a compact nonsingular real algebraic variety. A cohomology class in is said to be algebraic if the homology class Poincaré dual to it can be represented by a Zariski closed subvariety of of codimension , cf. [4, 7]. The set of all algebraic cohomology classes in forms a subgroup. Obviously, the unique generator of is an algebraic cohomology class.
The real algebraic varieties and are canonically biregularly isomorphic and will be identified. For any positive integer , let be the -fold product of . Clearly,
[TABLE]
We will regard as a subset of .
Fix an integer , and let be a point in . Let be the homology class in represented by the submanifold . Set
[TABLE]
where stands for the Kronecker product. Let be the canonical projection and let be a map of topological degree . For the map , we have
[TABLE]
Since the normal bundle to in is trivial and is the boundary of a compact manifold with boundary, it follows from [24, Proposition 2.5, Theorem 2.6] that there exist a nonsingular real algebraic variety and a diffeomorphism with
[TABLE]
The map
[TABLE]
is of class , and
[TABLE]
By [59, Propositions 7.2 and 7.7], is not homotopic to any stratified-regular map. In particular, cannot be approximated by stratified-regular maps, which is of interest in view of Theorems 1.3 and 1.5. Also, by Proposition 2.13, is not homotopic to any piecewise-regular map of class ; thus the case cannot be included in Theorem 1.6.
Furthermore, according to [16, Proposition 1.2], as above can be assumed to be a Zariski closed subvariety of that is obtained from via an arbitrarily small isotopy.
From a different viewpoint, strengthening of Theorems 1.3, 1.5 and 1.6 might be possible. Given a compact nonsingular real algebraic variety and two positive integers and , it remains an open question whether every continuous map from into or can be approximated by piecewise-regular maps of class .
Example 1.8**.**
Stratified-regular maps often have better approximation and homotopy properties than regular ones. For instance, if is a compact nonsingular real algebraic variety of dimension , then every continuous map from into can be approximated by stratified-regular maps, cf. [53, Corollary 1.3]. On the other hand, if is even, then each regular map from into is null homotopic, cf. [9] or [7, Theorem 13.5.1].
Piecewise-regular maps are not always more flexible than regular maps.
Example 1.9**.**
Let be the Fermat curve of degree in the real projective plane ,
[TABLE]
Clearly, can be identified with . If , then every piecewise-regular map from into is constant. This claim holds since, by virtue of the HurwitzâRiemann theorem [31, p. 140], every rational map from into is constant.
It would be of interest to decide whether or not counterparts of Theorems 1.3 and 1.5 hold for maps with values in an arbitrary rational nonsingular real algebraic variety.
We have already indicated how the present paper is organized. It should be added that Section 3 contains some preliminary technical results.
Henceforth, the following notation will be frequently used.
Notation 1.10**.**
For any function defined on some set , we put
[TABLE]
2 General properties of piecewise-regular maps
We first deal with regular maps in the sense of Definition 1.1.
Lemma 2.1**.**
Let be a real algebraic variety, a Zariski closed subvariety, and
[TABLE]
a map defined on some subset . Then the following conditions are equivalent:
- (a)
The map is regular. 2. (b)
Each component function is regular, .
Proof.
It is clear that (a) implies (b).
Suppose that (b) holds, and let be the Zariski closure of in . We can find a Zariski open neighborhood of and regular functions such that for . The regular map is continuous in the Zariski topology and . Hence , which implies (a). â
Regular functions can be characterized as follows.
Lemma 2.2**.**
Let be a Zariski closed subvariety, and a function defined on some subset . Then the following conditions are equivalent:
- (a)
The function is regular. 2. (b)
For each point there exist a Zariski open neighborhood of and a regular function such that on . 3. (c)
There exist two regular functions such that and on . 4. (d)
For each point there exist a Zariski open neighborhood of and two polynomial functions such that and on . 5. (e)
There exist two polynomial functions such that and on .
Proof.
It readily follows that
[TABLE]
Suppose that (d) holds. For each point , pick a polynomial function with . Since the Zariski topology on is Noetherian, we can find a finite subset such that
[TABLE]
Set , , , and
[TABLE]
Then and on . Consequently, on , hence (e) holds. The proof is complete. â
AÂ filtration of a real algebraic variety is a finite sequence of Zariski closed subvarieties of satisfying
[TABLE]
We allow for some . Note that is a stratification of .
The following is a generalization of [59, Proposition 2.2].
Proposition 2.3**.**
Let , be real algebraic varieties, and a map defined on some subset . Then the following conditions are equivalent:
- (a)
There exists a stratification of such that for every stratum the restriction of to is a regular map. 2. (b)
There exists a filtration of such that for every stratum the restriction of to is a regular map. 3. (c)
For every Zariski closed subvariety , there exists a Zariski open dense subset such that the restriction of to is a regular map.
*In particular, the map is stratified-regular if and only if it is continuous and satisfies the equivalent conditions *(a), (b), (c).
Proof.
It is clear that (b) implies (a).
Suppose that (a) holds, and let be an irreducible Zariski closed subvariety. We can find a stratum such that the intersection is nonempty and Zariski open (hence Zariski dense) subset of . Thus (c) holds for each irreducible . It immediately follows that (c) holds in the general case.
Now suppose that (c) is satisfied. Set . Making use of (c) with , we find a Zariski closed nowhere dense subvariety such that the restriction of to is a regular map. Note that . We repeat this construction with to get , and so on. This process terminates after finitely many steps with , which proves (b). â
Let , be real algebraic varieties, some subset, and the Zariski closure of in . We say that a map is rational if there exists a Zariski open dense subset such that the restriction of to is a regular map (no condition on the restriction of to is imposed).
In view of Proposition 2.3, each stratified-regular map is continuous rational. On the other hand, if the set of singular points of is nonempty, then it can happen that a function from into is continuous rational but it is not stratified-regular, cf. [43, Example 2]. However, the following holds.
Proposition 2.4**.**
Let , be real algebraic varieties, and a map defined on an open subset . Then the following conditions are equivalent:
- (a)
The map is stratified-regular. 2. (b)
The map is continuous and rational.
Proof.
We may assume that is a Zariski closed subvariety. Hence, by Lemma 2.1, the proof is reduced to the case , which follows from [42, Propostion 4.2] (a variant of [43, Proposition 8]). â
It is worthwhile to make a comment on the terminology used in different papers.
Remark 2.5**.**
In view of Proposition 2.4, reading papers [48, 50, 52, 53, 56, 58] one can substitute everywhere stratified-regular maps for continuous rational maps. It follows from Proposition 2.3 that stratified-regular functions coincide with continuous hereditarily rational functions studied in [42, 43]. Furthermore, as explained in [29, 59, 64], stratified-regular maps defined on a constructible subset of a real algebraic variety are identical with reguluous maps.
We now present a counterpart of Proposition 2.3 for piecewise-regular maps.
Proposition 2.6**.**
Let , be real algebraic varieties, and a map defined on some subset . Then the following conditions are equivalent:
- (a)
There exists a stratification of such that for every stratum the restriction of to each connected component of is a regular map. 2. (b)
There exists a filtration of such that for every stratum the restriction of to each connected component of is a regular map. 3. (c)
For every Zariski closed subvariety there exists a Zariski open dense subset such that the restriction of to each connected component of is a regular map.
*In particular, the map is piecewise-regular if and only if it is continuous and satisfies the equivalent conditions *(a), (b), (c).
Proof.
One can repeat the proof of Proposition 2.3 with only minor modifications. â
We also have the following characterization of piecewise-regular maps.
Proposition 2.7**.**
Let , be real algebraic varieties, and a continuous map defined on some subset . Then the following conditions are equivalent:
- (a)
The map is piecewise-regular. 2. (b)
There exists a stratification of such that for every stratum the restriction is a piecewise-regular map.
Proof.
It is clear that (a) implies (b).
Suppose that (b) holds for some stratification of . For each stratum there exists a stratification of such that for every stratum the restriction of to each connected component of is a regular map. Note that
[TABLE]
is a stratification of , and the map is piecewise -regular. Thus (b) implies (a), as required. â
Given a real algebraic variety , a subset is said to be a nonsingular algebraic arc if its Zariski closure in is an algebraic curve (that is, ), , and is homeomorphic to .
Proposition 2.8**.**
Let , be real algebraic varieties, and a continuous map defined on a semialgebraic subset . Then the following conditions are equivalent:
- (a)
The map is piecewise-regular. 2. (b)
There exists a stratification of such that for every stratum and every nonsingular algebraic arc in , with , the restriction is a regular map.
Proof.
We may assume that is a Zariski closed subvariety. Hence, by Lemma 2.1, the proof is reduced to the case .
It is clear that (a) implies (b).
Suppose that (b) holds for some stratification of . In view of [42, Proposition 3.5], for every stratum , the restriction is a piecewise-regular map. Consequently, by Proposition 2.7, is a piecewise-regular map. Hence (b) implies (a), as required. â
Piecewise-regular maps can be characterized among semialgebraic maps as follows.
Theorem 2.9**.**
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. 2. (b)
For every nonsingular algebraic arc in , with , the restriction is a piecewise-regular map. 3. (c)
For every nonsingular algebraic arc in , with , there exists a nonemtpy open subset such that the restriction is a regular map.
Proof.
As in the proof of Proposition 2.8, we may assume that .
Evidently, (a) implies (b), and (b) implies (c).
Suppose that (c) holds, and let be a Zariski closed subvariety. Let be the Zariski closure of in . By Lemma 2.11 below (with replaced by ), there exists a Zariski open dense subset such that the restriction of to each connected component of is a regular function. Note that is a Zariski open dense subset of , and
[TABLE]
Hence, in view of Proposition 2.6, condition (a) holds. â
For background on Nash manifolds and Nash functions we refer to [7]. The following variant of [42, Propositon 2.5] will be useful in the proof of Lemma 2.11.
Lemma 2.10**.**
Let be a connected Nash submanifold, and a Nash function. Assume that for every nonsingular algebraic arc in , with , there exists a nonempty open subset such that the restriction is a regular function. Then is a rational function.
Proof.
Let be the Zarsiki closure of in . Note that is irreducible. Furthermore, the Zariski closure of the graph of in is also irreducible. Complexifying these data, we complete the proof arguing as in [42, Propostion 2.5]. â
In the proof of Theorem 2.9 we used the following.
Lemma 2.11**.**
Let be a real algebraic variety, a semialgebraic set, and a semialgebraic function. Let be the Zariski closure of in . Assume that for every nonsingular algebraic arc in , with , there exists a nonempty open subset such that the restriction is a regular function. Then there exists a Zariski open dense subset such that the restriction of to each connected component of is a regular function.
Proof.
We may assume that is a Zariski closed subvariety. Set , and let be the interior of in . Then is a semialgebraic subset of whose Zariski closure is Zariski nowhere dense in , cf. [7, Chapter 2]. Furthermore, there exists a Zariski closed and Zariski nowhere dense subvariety such that
[TABLE]
and the restriction of to each connected component of is a Nash function. Since is a semialgebraic set, it has finitely many connected components. Hence, in view of Lemma 2.10, there exists a Zariski open dense subset which has the required properties. â
We next deal with piecewise-regular maps of class . Initially, we consider functions on nonsingular real algebraic arcs.
Lemma 2.12**.**
Let be a real algebraic curve, a nonsingular real algebraic arc, and a piecewise-regular function of class . Then is a regular function.
Proof.
The function is analytic, being semialgebraic and of class , cf. [7, Proposition 8.1.8].
By definition of piecewise-regular, we can find a Zariski open dense subset , a regular function and a nonempty open subset such that . Regarding as a subset of , we get a regular map with . Hence by the identity principle for analytic maps. Consequently, is a regular function. â
Lemma 2.12 can be generalized as follows.
Proposition 2.13**.**
Let , be real algebraic varieties, and a map defined on an open subset . Then the following conditions are equivalent:
- (a)
The map is piecewise-regular and of class . 2. (b)
The restriction of to each connected component of is a regular map.
Proof.
As in the proof of Proposition 2.8, we may assume that .
Suppose that (a) holds and let be a connected component of . For each nonsingular algebraic arc in , with , the restriction is a regular function by Lemma 2.12. Hence, in view of [42, Theorem 2.4], is a rational function. Consequently, is a regular function according to [48, Proposition 2.1].
It is clear that (b) implies (a). â
We also have the following variant of Proposition 2.13.
Proposition 2.14**.**
Let , be real algebraic varieties, and a map defined on an open subset . Then the following conditions are equivalent:
- (a)
The map is piecewise-regular, and for every nonsingular algebraic arc in , with , the restriction is of class . 2. (b)
The restriction of to each connected component of is a stratified-regular map.
Proof.
As in the proof of Proposition 2.8, we may assume that .
Suppose that (a) holds, and let be a connected component of . For each nonsingular algebraic arc in , with , the restriction is a regular function by Lemma 2.12. Hence, in view of [42, Theorem 2.4], is a rational function. Consequently, is a stratified-regular function according to Proposition 2.4.
It is clear that (b) implies (a). â
3 Functions on a simplex
This section is of a technical nature. Our main goal is Lemma 3.7, which is needed in Sections 4 and 6. For the sake of clarity, we begin with some preliminary facts.
Lemma 3.1**.**
Let be a real algebraic variety, a Zariski closed subvariety, and a regular function. Then there exists a regular function such that .
Proof.
By Lemma 2.1, there exist regular functions such that and on . Pick a regular function with . Then the function has the required properties. â
For any real algebraic variety , we let denote the ring of real-valued regular functions on . If is a Zariski closed subvariety, then the ideal
[TABLE]
of is called the ideal of in .
Lemma 3.2**.**
Let be a real algebraic variety, and , Zariski closed subvarieties of for which
[TABLE]
Let be a function such that the restrictions and are regular functions. Then is a regular function.
Proof.
By Lemma 3.1, there exists a regular function with for . Since , we have , where . Hence is a regular function on whose restriction to is equal to . Consequently, is a regular function. â
Let be a nonsingular real algebraic variety, and a Zariski closed subvariety. We say that is a simple normal crossing hypersurface if for each point there exist a Zariski open neighborhood of and local coordinates on (a regular system of parameters at ) such that the intersection of each irreducible component of with is given by the equation for a suitable . In particular, if is a simple normal crossing hypersurface, then each irreducible component of is nonsingular of codimension .
Lemma 3.3**.**
Let be a nonsingular real algebraic variety, a simple normal crossing hypersurface, and a function whose restriction to each irreducible component of is a regular function. Then is a regular function.
Proof.
We use induction on the number of irreducible components of . The case is obvious. Suppose that . Let be an irreducible component of , and let be the union of the remaining irreducible components. The restriction is a regular function by assumption, whereas the restriction is a regular function by the induction hypothesis.
Pick a point . It suffices to find a Zariski open neighborhood of such that is a regular function. If is small enough, there exist local coordinates on such that the ideal is generated by , and the ideal is generated by the product for some with . Note that the ideal is generated by and ; in other words,
[TABLE]
Hence is a regular function in view of Lemma 3.2. â
We give next the following variant of Lemma 3.1.
Lemma 3.4**.**
Let be a real algebraic variety, some subsets of , and a regular function. Assume that , where is the Zariski closure of in . Then there exists a regular function , defined on a Zariski open neighborhood of in , such that .
Proof.
By Lemma 2.2, there exist regular functions such that and on . Pick a regular function with . Set and on . Then has the required properties. â
Notation 3.5**.**
By a simplex in we always mean a closed geometric simplex. For any finite (geometric) simplicial complex in , we write for the union of all simplices in ; thus is a compact polyhedron. We denote by the -skeleton of .
If is a -simplex, then stands for the simplicial complex which consists of all faces of of dimension at most . Clearly, is the union of all -dimensional faces of . The Zariski closure of in , denoted by , is an affine subspace of dimension .
Lemma 3.6**.**
Let be a -simplex and let be a function such that the restriction is a regular function for every -simplex . Then there exists a regular function with .
Proof.
Let be all the -dimensional faces of . We set and for . Obviously, and . By Lemma 3.4, there exists a Zariski open neighborhood of and a regular function with for . In particular, on for all , . Since is the Zariski closure of in , we get on .
Set and . Then is a simple normal crossing hypersurface in . Define a function by for . By Lemma 3.3, is a regular function. In view of Lemma 3.1, there exists a regular function with . The function has the required properties. â
We need the following approximation result for functions defined on a simplex.
Lemma 3.7**.**
Let be a -simplex and let be a continuous function such that the restriction is a regular function for every -simplex . Then, for every , there exists a regular function satisfying
[TABLE]
and .
Proof.
According to Lemma 3.6, there exists a regular function with . By replacing with , the proof is reduced to the case .
Let be all the -dimensional faces of . We set and for . The union is a simple normal crossing hypersurface in . The function on which is equal to on and identically equal to [math] on is continuous. Hence, by Tietzeâs extension theorem, there exists a continuous function with and .
Fix . Note that there exists a function satisfying
[TABLE]
and . Indeed, by the Whitney approximation theorem [65, Theorem 10.16], one can find a function for which
[TABLE]
Since , the set is an open neighborhood of in . If is a function with and support contained in , then the function has the required properties.
Denote by the ring of real-valued functions on . One readily sees that the ideal of all functions vanishing on is generated by polynomial functions, say, (alternatively, one can invoke a much more general result [79, p. 52, Proposition 1]). Consequently, can be written in the form
[TABLE]
where the are functions on . Let
[TABLE]
By the Weierstrass approximation theorem, there exists a polynomial function satisfying
[TABLE]
For , we have
[TABLE]
and . We complete the proof setting . â
4 Piecewise-regular maps into Grassmannians
The role of Subsections 4.A and 4.B is to review some notation and terminology.
4.A Inner product and matrices
As in Section 1, we let denote , or . The -vector space is endowed with the standard inner product
[TABLE]
given by
[TABLE]
where stands for the conjugate of in .
Let , or simply if , denote the set of all -by- matrices with entries in . For any matrix , the corresponding -linear transformation is given by
[TABLE]
(recall that we always consider left -vector spaces). We will identify with and write
[TABLE]
If , then we define the product by
[TABLE]
This convention implies that .
We regard as a real algebraic variety. If , then the subset
[TABLE]
of all matrices with linearly independent columns is Zariski open. Furthermore, the map
[TABLE]
is a regular map, as is immediately seen by using the standard charts on .
4.B Vector bundles
For any topological -vector bundle on a topological space , we denote by its total space and by the bundle projection. The fiber of over a point is .
Given a nonnegative integer , we let denote the standard product -vector bundle on with total space . Any morphism of topological -vector bundles is of the form
[TABLE]
where is a uniquely determined map, called the matrix representation of . Obviously, is a continuous map.
If is a topological -vector subbundle of , then , where is the orthogonal complement of with respect to the standard inner product on . The orthogonal projection onto is a topological morphism of -vector bundles.
We will also consider algebraic vector bundles on a real algebraic variety . The product will be regarded as a real algebraic variety. By an algebraic -vector bundle on we mean an algebraic -vector subbundle of for some (cf. [7, Chapters 12 and 13] and [35, 36] for various characterizations of algebraic -vector bundles).
If is an algebraic morphism, then the matrix representation
[TABLE]
of is a regular map.
If is an algebraic -vector subbundle of , then its orthogonal complement is also an algebraic -vector subbundle, and the orthogonal projection onto is an algebraic morphism.
The tautological -vector bundle on is an algebraic -vector subbundle of .
Lemma 4.1**.**
Let be a real algebraic variety, and a continuous map defined on some subset . Then the map , where is the orthogonal projection onto for all , is continuous. Furthermore, if the map is regular, then so is the map .
Proof.
We regard the pullback as a topological -vector subbundle of , hence
[TABLE]
It follows that is the matrix representation of the orthogonal projection onto . Consequently, is a continuous map.
Now, suppose that is a regular map. It suffices to consider the case where is a Zariski locally closed subvariety of . Then is an algebraic -vector subbundle of , hence the argument above shows that is a regular map. â
4.C Maps into Grassmannians
We can now prove the following variant of Lemma 3.7.
Lemma 4.2**.**
Let be a -simplex and let be a continuous map such that the restriction is a regular map for every -simplex . Then, for each neighborhood of , there exists a regular map such that and .
Proof.
Consider the map , where is the orthogonal projection onto for all . By Lemma 4.1, is a continuous map and the restriction is a regular map for every -simplex .
We regard the pullback as a topological -vector subbundle of . Since  is topologically trivial, there exists an injective topological morphism whose image is equal to . Let be the matrix representation of . Then is a continuous map and
[TABLE]
By the Weierstrass approximation theorem, there exists a regular map arbitrarily close to . Define by
[TABLE]
Then is a continuous map, close to , such that the restriction is a regular map for every -simplex . Hence, according to Lemma 3.7, there exists a regular map , arbitrarily close to , with . In particular, the -linear transformation is injective for all . In other words, . Thus
[TABLE]
is a well-defined regular map, close to . We may assume that . Furthermore, since . â
An important consequence of Lemma 4.2 is the following.
Proposition 4.3**.**
Let be a finite simplicial complex in and let be a continuous map. Then, for each open neighborhood of , there exists a continuous map such that and the restriction is a regular map for every simplex .
Proof.
We use induction on . The case is obvious. Suppose now that . By the induction hypothesis, there exists a continuous map , arbitrarily close to , such that the restriction is a regular map for every simplex . It readily follows that has a continuous extension that belongs to . By Lemma 4.2, for every -simplex , there exists a regular map close to and such that . Then the map , defined by and for every -simplex , has all the required properties. â
In what follows, we will use stratifications constructed in a fairly simple way. Any finite simplicial complex in gives rise to a filtration
[TABLE]
of , where , , and is the union of the for all simplices of dimension at most with . Here, as in Notation 3.5, stands for the Zariski closure of . Setting
[TABLE]
we obtain a stratification of . More generally, if is a Zariski closed subvariety, then the collection
[TABLE]
is a stratification of , which is said to be induced by . Obviously,
[TABLE]
Lemma 4.4**.**
Let be a finite simplicial complex in , a Zariski closed subvariety, and an arbitrary subset. Then, for every stratum , each connected component of is contained in some simplex .
Proof.
It suffices to consider the case . If , then each connected component of is contained in a connected component of , which in turn is contained in some simplex by construction of . â
We are now ready to prove the first two theorems announced in Section 1.
Proof of Theorem 1.3.
Let be a positive integer. We may assume that and are Zariski closed subvarieties.
Consider a continuous map . By Tietzeâs extension theorem, there exists a continuous map with . Let be a tubular neighborhood of in . Then is a neighborhood of , and , given by for , is a continuous extension of . Since is a compact set, we can find a finite simplicial complex in with . In view of Proposition 4.3, there exists a continuous map , arbitrarily close to , such that the restriction is a regular map for every simplex . According to Lemma 4.4, the restriction is a piecewise -regular map. The proof is complete since is close to . â
Proof of Theorem 1.4.
It can be assumed that is a Zariski closed subvariety, hence . By Borsukâs theorem [27, p. 537], is a retract of some neighborhood . We can find a finite simplicial complex in with . Thus there exists a retraction .
We claim that the induced stratification of has all the required properties. Indeed, let be a positive integer and let be a continuous map. Then is a continuous extension of . By Proposition 4.3, there exists a continuous map , arbitrarily close to , such that the restriction is a regular map for every simplex . In view of Lemma 4.4, the restriction is a piecewise -regular map. This completes the proof since is close to . â
We also have the following variant of Theorem 1.4.
Theorem 4.5**.**
Let be a compact subset, a neighborhood of , a homeomorphism, and is a finite simplicial complex in . Assume that is a retract of and
[TABLE]
Then, for each positive integer , every continuous map from into can be approximated by piecewise -regular maps.
Proof.
Let be a retraction. Then , given by
[TABLE]
is a well-defined retraction.
Let be a positive integer and let be a continuous map. Since is a continuous extension of , we complete the proof as in the case of Theorem 1.4. â
Theorem 4.5 can be illustrated by revisiting Example 1.7.
Example 4.6**.**
Fix an integer , and consider as a subset of . Note that is a retract of . Let be a finite simplicial complex in satisfying
[TABLE]
According to Example 1.7, we can find a nonsingular Zariski closed subvariety , a diffeomorphism , and a map
[TABLE]
such that and is not homotopic to any stratified-regular map.
By Theorem 4.5, for each positive integer , every continuous map from into can be approximated by piecewise -regular maps. In particular, this is the case for the map .
5 Piecewise-algebraic vector bundles
Comparison of algebraic and topological vector bundles on a given real algebraic variety is a challenging problem, cf. [3â7,12,13,15,22,23,26,28,32,40,45â47,75,77,78]. Unless the variety is quite exceptional, one can find a topological vector bundle on that is not topologically isomorphic to any algebraic vector bundle. On the other hand, in some sense, algebraization of topological vector bundles is possible, cf. [3, 47, 67, 68, 77, 78]. In this section we look at the algebraization problem from a new perspective. The main results are Theorems 5.10 and 5.11, derived from Theorems 1.3 and 1.4, respectively.
We will use freely notation introduced in Section 4. Moreover, given a topological space , a subspace , and a topological morphism of topological -vector bundles on , we let denote the restriction morphism defined by for all .
We first generalize the definition of algebraic vector bundle.
Definition 5.1**.**
Let be a real algebraic variety, some 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 . In that case, is also said to be an algebraic -vector subbundle of . The pair is called an algebraic extension of . In particular, is an algebraic -vector bundle on .
If and are algebraic -vector bundles on , then an algebraic morphism is a topological morphism such that there exist algebraic extensions , of , , respectively, and an algebraic morphism with .
Algebraic -vector bundles on , together with algebraic morphisms, form a category.
Lemma 5.2**.**
Let be a real algebraic variety, some subset, and a bijective algebraic morphism of algebraic -vector bundles on . Then is an algebraic isomorphism.
Proof.
Let , , and be as in Definition 5.1. Shrinking if necessary, we may assume that the algebraic morphism is bijective. Thus the proof is reduced to the case where  is a Zariski locally closed subvariety of .
Our goal is to prove that the inverse map is a regular map. The problem is local for the Zariski topology on , so it suffices to consider . Let be the matrix representation of . Then is a regular map and
[TABLE]
Consequently, is a regular map, as required. â
Lemma 5.3**.**
Let be a real algebraic variety, some subset, and an algebraic -vector subbundle of . Then the orthogonal complement of is also an algebraic -vector subbundle of , and the orthogonal projection is an algebraic morphism.
Proof.
This is a standard fact if is a Zariski locally closed subvariety of . The general case follows immediately. â
We denote by the algebraic -vector subbundle of whose restriction to is for .
Given a topological space and a continuous map , we regard the pullback as a topological -vector subbundle of ; thus
[TABLE]
Conversely, if is a topological -vector subbundle of , then the map , defined by
[TABLE]
is continuous and . We call the classifying map for . Note that the classifying map for is .
Lemma 5.4**.**
Let be a real algebraic variety, some subset, and a topological -vector subbundle of . Then the following conditions are equivalent:
- (a)
* is an algebraic -vector subbundle of .* 2. (b)
The classifying map for is regular.
Proof.
This is well-known if is a Zariski locally closed subvariety of . The general case follows immediately. â
Next we introduce the central notion of this section.
Definition 5.5**.**
Let be a real algebraic variety, some subset, and a stratification of .
An -algebraic (resp. a piecewise -algebraic) -vector bundle on is a topological -vector subbundle of , for some , such that for every stratum the restriction is an algebraic -vector subbundle of (resp. for every stratum and each connected component of the restriction is an algebraic -vector subbundle of ). In that case, is also said to be an -algebraic (resp. a piecewise -algebraic) -vector subbundle of .
If and are -algebraic (resp. piecewise -algebraic) -vector bundles on , then an -algebraic (resp. a piecewise -algebraic) morphism is a topological morphism such that for every stratum the restriction is an algebraic morphism (resp. for every stratum and each connected component of the restriction is an algebraic morphism).
A stratified-algebraic (resp. a piecewise-algebraic) -vector bundle on is a -algebraic (resp. a piecewise -algebraic) -vector bundle on for some stratification of .
If and are stratified-algebraic (resp. piecewise-algebraic) -vector bundles on , then a stratified-algebraic (resp. a piecewise-algebraic) morphism is a -algebraic (resp. a piecewise -algebraic) morphism for some stratification of such that both and are -algebraic (resp. piecewise -algebraic) -vector bundles on .
It is clear that -algebraic (resp. piecewise -algebraic) -vector bundles on , together with -algebraic (resp. piecewise -algebraic) morphisms, form a category. Similarly, stratified-algebraic (resp. piecewise-algebraic) -vector bundles on , together with stratified-algebraic (resp. piecewise-algebraic) morphisms, form a category.
In a somewhat less general context, -algebraic and stratified-algebraic -vector bundles are thoroughly investigated in [55, 57, 59, 61, 64].
Proposition 5.6**.**
Let be a real algebraic variety, some subset, a stratification of , and a topological -vector subbundle of for some . Then the following conditions are equivalent:
- (a)
* is an -algebraic (resp. a piecewise -algebraic) -vector subbundle of .* 2. (b)
The classifying map for is -regular (resp. piecewise -regular).
Proof.
This follows from Lemma 5.4. â
As a direct consequence, we obtain the following.
Corollary 5.7**.**
Let be a real algebraic variety, some subset, and a topological -vector subbundle of for some . Then the following conditions are equivalent:
- (a)
* is a stratified-algebraic (resp. a piecewise-algebraic) -vector subbundle of .* 2. (b)
The classifying map for is stratified-regular (resp. piecewise-regular).
Vector bundles introduced in Definition 5.5 can be compared with topological vector bundles.
Proposition 5.8**.**
Let be a real algebraic variety, a compact subset, and a stratification of . Let and be -algebraic (resp. piecewise -algebraic) -vector bundles on that are topologically isomorphic. Then and are also isomorphic in the category of -algebraic (resp. piecewise -algebraic) -vector bundles on .
Proof.
We consider explicitly only the piecewise -algebraic case, the -algebraic one being completely analogous.
Thus, (resp. ) is a piecewise -algebraic -vector subbundle of (resp. ) for some positive integer (resp. ). Since and , there exists a topological morphism which transforms onto . Let be the matrix representation of . By the Weierstrass approximation theorem, we can find a regular map that is close to . Then , defined by
[TABLE]
is an algebraic morphism.
In view of Lemma 5.3, the orthogonal projection onto is a piecewise -algebraic morphism. Hence is a piecewise -algebraic morphism which transforms onto . Consequently, the morphism determined by is bijective and piecewise -algebraic. It follows from Lemma 5.2 that is a piecewise -algebraic isomorphism. â
Proposition 5.8 implies immediately the following.
Corollary 5.9**.**
Let be a real algebraic variety, and a compact subset. Let and be stratified-algebraic (resp. piecewise-algebraic) -vector bundles on that are topologically isomorphic. Then and are also isomorphic in the category of stratified-algebraic (resp. piecewise-algebraic) -vector bundles on .
As an application of Theorem 1.3, we obtain the following result.
Theorem 5.10**.**
Let be a real algebraic variety and let be a compact subset. Then each topological -vector bundle on is topologically isomorphic to a piecewise-algebraic -vector bundle on , which is uniquely determined up to piecewise-algebraic isomorphism.
Proof.
Let be a topological -vector bundle on . Since is compact, one can find a positive integer and a continuous map such that is topologically isomorphic to the pullback , cf. [37, Chapter 3, Proposition 5.8]. According to Theorem 1.3, is homotopic to a piecewise-regular map , hence is topologically isomorphic to the pullback , cf. [37, Chapter 3, Theorem 4.7]. By Corollary 5.7, is a piecewise-algebraic -vector bundle on . The proof is complete in view of Corollary 5.9. â
In a similar way, we can derive from Theorem 1.4 the next result.
Theorem 5.11**.**
Let be a real algebraic variety and let be a compact locally contractible subset. Then there exists a stratification of such that each topological -vector bundle on is topologically isomorphic to a piecewise -algebraic -vector bundle on , which is uniquely determined up to piecewise -algebraic isomorphism.
Proof.
According to Theorem 1.4, there exists a stratification of such that, for each positive integer , every continuous map from into is homotopic to a piecewise -regular map.
Let be a topological -vector bundle on . One can find a positive integer and a continuous map such that is topologically isomorphic to the pullback . Hence is topologically isomorphic to the pullback , where is a piecewise -regular map homotopic to . By Proposition 5.6, is a piecewise -algebraic -vector bundle on . The proof is complete in view of Proposition 5.8. â
In Theorems 5.10 and 5.11, piecewise-algebraic and piecewise -algebraic cannot be replaced by stratified-algebraic and -algebraic, respectively.
Example 5.12**.**
Fix an integer , and let and be as in Example 1.7. We claim that the topological -vector bundle on is not topologically isomorphic to any stratified-algebraic -vector bundle. Indeed, write and to indicate that and are regarded as topological -vector bundles. Since , we get
[TABLE]
where stands for the th StiefelâWhitney class. The claim follows in view of [59, Propositions 7.3 and 7.7].
6 Piecewise-regular maps into spheres (approximation)
For maps with values in , we have the following counterpart of Proposition 4.3.
Proposition 6.1**.**
Let be a finite simplicial complex in and let be a continuous map. Assume that for every simplex . Then, for each open neighborhood of , there exists a continuous map such that and the restriction is a regular map for every simplex .
Proof.
We use induction on . The case is obvious. Suppose now that . By the induction hypothesis, there exists a continuous map , arbitrarily close to , such that the restriction is a regular map for every simplex . It readily follows that can be extended to a continuous map that belongs to and satisfies for every simplex . Since with one point removed is biregularly isomorphic to , it follows from Lemma 3.7 that for every -simplex , there exists a regular map close to and such that . Then the map , defined by and for every -simplex , has all the required properties. â
In what follows we will make use of Lemma 4.4.
Proof of Theorem 1.5.
Let be a positive integer and let be a continuous map. We may assume that is a Zariski closed subvariety, hence . By Tietzeâs extension theorem, there exists a continuous map with . Denoting by the radial projection, we see that is a neighborhood of , and , given by for , is a continuous extension of .
We can find a finite simplicial complex in with . By replacing with a suitable iterated barycentric subdivision of , we get for every simplex . In view of Proposition 6.1, we can find a continuous map , arbitrarily close to , such that the restriction is a regular map for every simplex . According to Lemma 4.4, the restriction is a piecewise -regular map. The proof is complete since is close to . â
It is not clear whether there is a counterpart of Theorem 1.4 for maps into spheres.
Problem 6.2**.**
Let be a real algebraic variety, a compact locally contractible subset, and a positive integer. Does there exist a stratification of such that every continuous map from into can be approximated by piecewise -regular maps?
In view of Theorem 1.4, the answer is affirmative if or since .
7 Piecewise-regular maps into spheres (homotopy)
The following lemma will be used in the proof of Theorem 1.6.
Lemma 7.1**.**
Let be a compact nonsingular real algebraic variety, a nonsingular Zariski closed subvariety with , and the union of some connected components of . Then there exists a closed tubular neighborhood of such that and the boundary of is a nonsingular Zariski closed subvariety of . Furthermore, for each nonnegative integer , there exists a function with the following properties:
- (i)
; 2. (ii)
the restrictions and are regular functions.
Proof.
Let be a closed tubular neighborhood of with . Note that the homology class in represented by the hypersurface is equal to [math]. Hence there exists a diffeomorphism , arbitrarily close in the topology to the identity map of , such that is a nonsingular Zariski closed subvariety of and for all , cf. [7, Theorem 12.4.11]. Replacing by , we may assume that is a nonsingular Zariski closed subvariety of .
Let and be real-valued regular functions on with and . If is a nonnegative integer, then
[TABLE]
is a function that satisfies (i) and (ii). â
Our proof of Theorem 1.6 depends on the PontryaginâThom construction. Unless explicitly stated otherwise, all manifolds will be without boundary. Submanifolds will be closed subsets of the ambient manifold. The unit -sphere will be oriented as the boundary of the unit -disc. For any compact manifold there is a canonical one-to-one correspondence
[TABLE]
where is the set of all homotopy classes of continuous maps from into , and is the set of framed cobordism classes of framed submanifolds of of codimension , cf. [44, 71] for details. Given a continuous map , we denote by the element of corresponding to the homotopy class of .
It is convenient to introduce some notation related to this construction. A framed submanifold of of codimension is a pair , where is a codimension submanifold, and is a framing of the normal bundle to in . The normal space to in at is the quotient of the tangent spaces; thus is a basis for .
Given a continuous map and a point , suppose that for some open neighborhood of the restriction map is of class and transverse to . Choose a positively oriented basis for . Then is represented by the framed submanifold , where and is transformed onto by the isomorphism induced by the derivative for every .
Let be a map transverse to and let be the union of some connected components of . Then we obtain a framed submanifold of , where and is transformed onto the canonical basis for by the isomorphism induced by the derivative for every .
Proof of Theorem 1.6.
The case is obvious since then is null homotopic. Suppose that , and let be a framed submanifold of which represents . By a standard transversality argument, we obtain a map such that is transverse to , is the union of some connected components of , and is framed cobordant to . In view of the Weierstrass approximation theorem, there exists a regular map arbitrarily close to in the topology. Then is transverse to and is isotopic to , cf. [2, p. 51]. In particular, is framed cobordant to , where is the union of suitable connected components of . Furthermore, is a nonsingular Zariski closed subvariety. Set , and let , be as in Lemma 7.1 for some nonnegative integer . According to the Ćojasiewicz inequality [7, Corollary 2.6.7], we can find an open neighborhood of , a real number , and an integer such that
[TABLE]
where stands for the Euclidean norm on .
Set , , and let be the stereographic projection. Fix a nonnegative integer . If and are sufficiently large integers, then the map defined by
[TABLE]
is of class . Since is a biregular isomorphism, the restrictions and are regular maps. Consequently, is a stratification of , and the map is piecewise -regular.
It remains to prove that is homotopic to or, equivalently, . To this end, let be the basis for which corresponds to the canonical basis for via the isomorphism . Note that the restriction of to is a map transverse to and . It readily follows that is framed cobordant to . Hence is framed cobordant to , which implies that , as required. â
Remark 7.2**.**
As demonstrated above, if , then the stratification that appears in Theorem 1.6 can be chosen of the form
[TABLE]
where and are disjoint Zariski closed subvarieties of with and .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1]
- 2[2] R. Abraham and J. Robbin, Transversal Mappings and Flows, Benjamin, 1967.
- 3[3] R. Benedetti and A. Tognoli, On real algebraic vector bundles, Bull. Sci. Math. 104 (1980), no. 2, 89â112.
- 4[4] R. Benedetti and A. Tognoli, Remarks and counterexamples in the theory of real vector bundles and cycles, in: GĂ©omĂ©trie algĂ©brique rĂ©elle et formes quadratiques, Lecture Notes in Math. 959 (1982), Springer, 198â211.
- 5[5] M. Bilski, W. Kucharz, A. Valette and G. Valette, Vector bundles and regulous maps, Math. Z. 275 (2013), 403â418.
- 6[6] J. Bochnak, M. Buchner and W. Kucharz, Vector bundles over real algebraic varieties, K-Theory 3 (1989), 271â298. Erratum in K-Theory 4 (1990), 113.
- 7[7] J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Ergeb. Math. Grenzgeb. Vol. 36, Springer, 1989.
- 8[8] J. Bochnak and W. Kucharz, Algebraic approximation of mappings into spheres, Michigan Math. J. 34 (1987), 119â125.
