On covering dimension and sections of vector bundles
M. C. Crabb

TL;DR
This paper uses elementary topology and cohomology to unify and extend classical theorems in topology and combinatorics, including Lebesgue, Knaster-Kuratowski-Mazurkiewicz, and Tverberg theorems.
Contribution
It provides a new elementary approach to classical results in topology and combinatorics using mod 2 cohomology of real projective spaces.
Findings
Unified proofs of classical theorems in topology and combinatorics
Extension of the topological central point theorem
New applications to Helly-Lovász, Bárány, and Tverberg results
Abstract
An elementary result in point-set topology is used, with knowledge of the mod cohomology of real projective spaces, to establish classical results of Lebesgue and Knaster-Kuratowski-Mazurkiewicz, as well as the topological central point theorem of Karasev, which is applied to deduce results of Helly-Lov\'asz, B\'ar\'any and Tverberg
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
TopicsAdvanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology
\UseRawInputEncoding
On covering dimension and sections of vector bundles
M. C. Crabb
Institute of Mathematics
University of Aberdeen
Aberdeen AB24 3UE
UK
Abstract.
An elementary result in point-set topology is used, with knowledge of the mod cohomology of real projective spaces, to establish classical results of Lebesgue and Knaster-Kuratowski-Mazurkiewicz, as well as the topological central point theorem of Karasev, which is applied to deduce results of Helly-Lovász, Bárány and Tverberg.
Key words and phrases:
vector bundle, covering dimension, Euler class
1991 Mathematics Subject Classification:
Primary 55R25, 54F45, 55M10, Secondary 55R40, 55M30
1. Introduction
Throughout this note, will be a compact Hausdorff topological space and , , () will be finite-dimensional real vector bundles over . We write .
Suppose that are closed subspaces covering and that, for each , the restriction of to admits a nowhere zero section, say. By Tietze’s theorem, extends to a section of on . Then is a nowhere zero section of . (And conversely, if admits a nowhere zero section , we may construct such closed subspaces by choosing an inner product on each and setting .) The main result of this paper, modelled on a cohomological lemma [7, Lemma 3.2] of Karasev and stated as Theorem 2.1, is in the same vein as this classical observation. The methods are from elementary point-set topology. As applications we derive in Sections 3 and 4, using ideas introduced by Karasev in [6, 7], a classical result of Lebesgue and Knaster-Kuratowski-Mazurkiewisz and the more recent topological central point theorem of Karasev with, as corollaries, results of Helly-Lovász and Bárány [2].
When we discuss Euler classes, we shall use representable cohomology as, for example, in [4, Section 8].
From the foregoing summary it should be clear that most of the ideas presented in this note derive from the paper [7] of Karasev. It is hoped, nevertheless, that the elementary approach taken here may have some conceptual advantages.
2. The principal result
Theorem 2.1**.**
Let , , be finite-dimensional real vector bundles over a compact Hausdorff topological space . Suppose that is a finite open cover of such that each point of lies in for at most indices .
Suppose that for each and there exists a section of with no zeros in . Then admits a global nowhere zero section.
Proof.
We begin with an argument from [9, Lemma 2.4]. Choose a partition of unity subordinate to the cover. For , define
[TABLE]
By assumption (and is non-empty). For a non-empty subset , we now set
[TABLE]
It is clearly an open subset of . Moreover, , so that the sets cover . If and are distinct subsets of with , then . (For there exist elements , with , and , with .)
Choose a partition of unity subordinate to the open cover of and, for each with , a section of which is nowhere zero on (possible because if ). We can then define a section of by
[TABLE]
Notice that, if for some with , then . For, if is a different subset with , .
The section of is nowhere zero. ∎
Corollary 2.2**.**
Suppose that is a finite closed cover of such that each point of lies in for at most indices .
Suppose that for each and the restriction of to admits a nowhere zero section. Then admits a global nowhere zero section.
Proof.
It is an elementary exercise to show that there is an open cover such that (i) , (ii) for each there exits a global section of with no zeros in , and (iii) each point of lies in at most of the sets . Then we can apply Theorem 2.1.
(Here are the details when . Consider a subset with . The open sets , , cover . Choose a partition of unity subordinate to this cover. Then we can define . By construction (because for any ) and for .
Every nowhere zero section of over the closed set extends to a global section of and such a section will be nowhere zero on an open neighbourhood of . So it is easy to choose in the intersection of the sets with and to satisfy (i) and (ii). And then, for any set with , we have .) ∎
3. A theorem of Lebesgue
If the mod cohomology Euler class of is non-zero, the zero-set of any section of is non-empty.
Proposition 3.1**.**
Let be a continuous map from to a compact Hausdorff space . Suppose that is a finite cover of by closed sets such that any point of lies in at most of the sets .
If the mod cohomology Euler class of is non-zero, then there exist and such that is non-empty for each section of .
Proof.
We take in Corollary 2.2. Since , does not admit a nowhere zero section. Hence there is a pair such that for every section of . ∎
For a finite set of cardinality , we write for the -dimensional real vector space of maps , for the -simplex of maps such that for all and , for the unit -sphere in of vectors with and for the -dimensional real projective space of lines in (where is non-zero). There is a surjective map
[TABLE]
defined by for . There is also an embedding
[TABLE]
defined by , such that the composition
[TABLE]
is the identity. The Hopf line bundle over , with fibre at , is denoted by .
The support, , of is the set of points such that . For an integer , , we write for the set of finite subsets of such that any two elements of have disjoint supports and . For , we write for the convex hull of ; it is a simplex of codimension in .
Lemma 3.2**.**
For , the simplex can be expressed as for some section of the vector bundle .
Proof.
Let be the codimension vector subspace of spanned by . The section of over given by the projection , for , has the property that . Choose some isomorphism to get the required section . ∎
Theorem 3.3**.**
Let be a family of finite sets with , where and , , are positive integers. Write . Suppose that is a finite closed cover of
[TABLE]
such that any point of lies in at most of the sets .
Then for some and the projection of to the th factor meets each of the codimension simplices for .
The Lebesgue theorem [7, Theorem 4.1] is the special case , , for all ; the case , is a result of Knaster, Kuratowski, Mazurkiewicz [7, Remark 2.2].
We follow the method of Karasev [7, Theorems 2.1 and 4.1].
Proof.
This is an immediate consequence of Proposition 3.1. Take to be the product
[TABLE]
and to be the multiple of the Hopf bundle over if . The Euler class is non-zero. ∎
The next lemma gives a way of checking that in the application of Proposition 3.1 is non-zero.
Lemma 3.4**.**
Suppose that is a closed subspace of a compact Hausdorff space and that is the restriction of a vector bundle over . Let be a real vector bundle over with the property that the restriction of to each connected component of the complement admits a nowhere zero section.
If the mod cohomology Euler class is non-zero, then is non-zero.
Proof.
Fix a Euclidean metric on . For each component of choose a continuous section of the sphere bundle , and choose a continuous function such that . Then one can define a continuous section of with zero-set by if , if .
Since is non-zero, the restriction of to is non-zero. See, for example, [4, Proposition 2.7]. ∎
This allows us to deduce Karasev’s strengthened KKM theorem [7, Theorem 2.1]).
Proposition 3.5**.**
Let be closed subsets of a simplex with vertex set of cardinality , where and are integers, such that any point of lies in at most of the sets . Then, either for some the subset meets each codimension simplex for , or some connected component of intersects every -dimensional simplex for .
Proof.
Considering , take to be and to be the restriction of . Let each be the restriction of , so that is the restriction of to . Take , so that is non-zero.
If, for each component of , written as the image of a component of , there is some simplex , where , such that , then Lemma 3.2 provides a nowhere zero section of over . By Lemma 3.4, is then non-zero, and we can apply Proposition 3.1 to deduce the existence of an such that meets each codimension simplex. ∎
4. Karasev’s topological central point theorem
An early result of the following type appears in [11, Lemma 3.1].
Proposition 4.1**.**
Let be a continuous map from to a compact Hausdorff space with covering dimension less than . Suppose that the mod cohomology class is non-zero.
Then there exists a point and such that for each section of .
Proof.
Suppose that for each point there exist sections of , , such that for each . Then the open sets , , cover . Since is compact with covering dimension , this open cover may be refined by a finite open cover such that each point of lies in at most of the sets .
Set . Then we may apply Theorem 2.1 to the open cover of to conclude that there exist and such that for every section of the zero-set meets . So for every section of . But for some . This contradiction completes the proof. ∎
Theorem 4.2**.**
Let be a family of finite sets with , where , are positive integers. Write . Suppose that
[TABLE]
is a continuous map to a compact Hausdorff space with covering dimension less than .
Then for some
[TABLE]
Karasev’s topological central point theorem, as in [6, Theorem 1.1] and [7, Theorem 5.1], is the special case .
Proof.
We apply Proposition 4.1 with and as in Theorem 3.3 and with equal to the composition of
[TABLE]
with . We recall that if , so that the Euler class is non-zero. ∎
As an application we prove a result of Helly-Lovász [2, Theorem 3.1].
Corollary 4.3**.**
Suppose that , , , , are convex subsets of a real vector space with the property that the intersection is non-empty for each .
If , then, for some , the intersection is non-empty.
Proof.
We apply Theorem 4.2 with , , . For each choose . Take to be the piecewise linear map
[TABLE]
We conclude from Theorem 4.2, noting that a codimension simplex in is a point, that there is some and such that, for each the vector can be written as
[TABLE]
where for each and so . Since each with lies in the convex set , we see that , as required. ∎
Bárány’s dual result [2, Theorem 2.1] (as formulated in [5, Theorem 3.1] and [8, Theorem 3]) can be obtained in a similar fashion.
Corollary 4.4**.**
Let be a non-empty compact convex subspace of a finite-dimensional real vector space . Suppose that are finite sets with , , and that are maps with the property that for each the convex hull of in is disjoint from .
If , then, for some , the convex hull of in is disjoint from .
Proof.
Choose a basepoint . We again apply Theorem 4.2 with , , and take to be the affine space of affine linear maps such that . Notice that the covering dimension of is equal to .
For each choose an affine linear map taking strictly positive values on and strictly negative values on . As in the proof of Helly’s theorem, take to be the piecewise linear map
[TABLE]
Theorem 4.2 provides some and such that, for each the affine linear map can be written as
[TABLE]
where , so that takes a strictly positive value at each and strictly negative values on . ∎
As observed by Sarkaria [10] (and expounded in [3]), Tverberg’s theorem is an easy consequence of Corollary 4.4. The following generalization, discussed in [3, Theorem 3.8] and due to Arocha, Bárány, Bracho, Fabila and Montejano [1], can be viewed as a coincidence theorem.
Corollary 4.5**.**
Let and be integers. For , , let be non-empty finite sets and be maps to a finite-dimensional real vector space satisfying the two conditions:
(i)* for each , there is a non-zero vector in that can be expressed, for each , as a linear combination with non-negative coefficients of the elements of ;*
(ii)* for each and , , the convex hull of is disjoint from .*
Then, if , there is a partition into non-empty subsets and a non-zero vector such that
[TABLE]
for some and for .
If, further, there is some affine hyperplane in that contains all the subsets but does not contain [math], then may be chosen in the hyperplane and then for each .
Proof.
Let be the quotient of by the subspace generated by and write for the coset of .
We take , , defined by for , and apply Corollary 4.4 with . By assumption, the convex hull of each in contains [math]. (Notice that, if , then if and only if .)
So there exist , , , and a non-zero such that . Take .
If there is a linear form taking the value on all , we can scale to arrange that , and then . ∎
The original Tverberg theorem is the case in which is a single point for all and is independent of .
Appendix A Cohomology
It is a classical result that, if is a closed cover of a compact Hausdorff space and, for each , is a mod cohomology class of that restricts to zero on , then the product is zero. Here is the corresponding version of Corollary 2.2, which was used by Karasev in the form [7, Lemma 3.2].
Theorem A.1**.**
Let be a compact Hausdorff space and let by classes in the mod cohomology of .
Suppose that is a finite closed cover of such that each point of lies in for at most indices and that for each and the restriction of to is zero.
Then .
Proof.
By the argument used in the proof of Corollary 2.2 one can manufacture an open cover such that each cohomology class is represented by a map which is null (not just null-homotopic) on . The construction in the proof of Theorem 2.1 gives an open cover indexed by the non-empty subsets of with . The map representing is null on the disjoint union of the sets with . Since these open sets cover , the product is represented by the null map, and so the cohomology class is zero. ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] J. L. Arocha, I. Bárány, J. Bracho, R. Fabila and L. Montejano, Very colorful theorems. Discrete Comput. Geom. 42 (2009), 142–154.
- 2[2] I. Bárány, A generalization of Carathéodory’s theorem. Discrete Math. 40 (1982), 141–152.
- 3[3] I. Bárány and P. Soberón, Tverberg’s theorem is 50 50 50 year old: a survey. Bull. Amer. Math. Soc. 55 (2018), 459–492.
- 4[4] M. C. Crabb and J. Jaworowski, Aspects of the Borsuk-Ulam theorem. J. Fixed Point Theory Appl. 13 (2013) 459–488.
- 5[5] A. F. Holmsen and R. Karasev, Colorful theorems for strong convexity. Proc. Amer. Math. Soc. 145 (2017), 2713–2726.
- 6[6] R. N. Karasev, A topological central point theorem. Top. Appl. 159 (2012), 864–868.
- 7[7] R. N. Karasev, Covering dimension using toric varieties. Top. Appl. 177 (2014), 59–65.
- 8[8] N. B. Mustafa and S. Ray, An optimal generalization of the Colorful Carathéodory theorem. Discrete Math. 339 (2016), 1300–1305.
