The Topological Transversal Tverberg theorem plus Constraints

The Topological Transversal Tverberg theorem plus Constraints

Pavle V. M. Blagojević Inst. Math., FU Berlin, Arnimallee 2, 14195 Berlin, Germany     Mat. Institut SANU, Knez Mihailova 36, 11001 Beograd, Serbia blagojevic@math.fu-berlin.de Aleksandra S. Dimitrijević Blagojević Mat. Institut SANU, Knez Mihailova 36, 11001 Beograd, Serbia aleksandra1973@gmail.com  and  Günter M. Ziegler Inst. Math., FU Berlin, Arnimallee 2, 14195 Berlin, Germany ziegler@math.fu-berlin.de
March 31, 2016; revised October 10, 2016
To appear in “Discrete and Intuitive Geometry – László Fejes Tóth 100 Festschrift” (G. Ambrus, I. Bárány, K. J. Böröczky, G. Fejes Tóth, J. Pach, eds.), Bolyai Society Mathematical Studies series, Springer
Abstract.

In this paper we use the strength of the constraint method in combination with a generalized Borsuk–Ulam type theorem and a cohomological intersection lemma to show how one can obtain many new topological transversal theorems of Tverberg type. In particular, we derive a topological generalized transversal Van Kampen–Flores theorem and a topological transversal weak colored Tverberg theorem.

The research by Pavle V. M. Blagojević leading to these results has received funding from DFG via Berlin Mathematical School. Also supported by the grant ON 174008 of the Serbian Ministry of Education and Science.
The research by Aleksandra Dimitrijević Blagojević leading to these results has received funding from the grant ON 174008 of the Serbian Ministry of Education and Science.
The research by Günter M. Ziegler received funding from DFG via the Research Training Group “Methods for Discrete Structures” and the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.”

1. Introduction

At the Symposium on Combinatorics and Geometry in Stockholm 1989, Helge Tverberg formulated the following conjecture that in a special case coincides with his famous 1966 result [Tverberg1966, Thm. 1].

Conjecture 1.1 (The transversal Tverberg conjecture).

Let

  • and be integers with ,

  • be integers, and

  • .

Then for every collection of sets with , there exist an -dimensional affine subspace of and pairwise disjoint subsets of , for , such that

For this conjecture is Tverberg’s well-known theorem. Tverberg and Vrećica published the full conjecture in 1993 [Tverberg1993]. They proved that it also holds for [Tverberg1993, Prop. 3]. For and arbitrary they verified the conjecture only in the following three cases: , , and [Tverberg1993, Prop. 1].

\includegraphics

[scale=1.3]Transversal-01.pdf

Figure 1. The transversal Tverberg conjecture for , and or .

The classical Tverberg theorem from 1966 was extended to a topological setting by Bárány, Shlosman, and Szűcs [Barany1981] in 1981. Similarly, it is natural to consider the following extension of the transversal Tverberg conjecture.

Conjecture 1.2 (The topological transversal Tverberg conjecture).

Let

  • and be integers with ,

  • be integers, and

  • .

Then for every collection of continuous maps there exist an -dimensional affine subspace of and pairwise disjoint faces such that

In 1999 using advanced methods of algebraic topology Živaljević [Zivaljevic1999, Thm. 4.8] proved this conjecture for and odd integers and being an odd prime. The topological transversal Tverberg conjecture was settled for by Vrećica [Vrecica2003, Thm. 2.2] in 2003. In 2007 Karasev [Karasev2007, Thm. 1] established the topological transversal Tverberg conjecture in the cases when integers are, not necessarily equal, powers of the same prime and the product is even.

In the same paper Karasev [Karasev2007] proved a colored topological transversal Tverberg’s theorem [Karasev2007, Thm. 5], which for coincides with the colored Tverberg theorem of Živaljević and Vrećica [Zivaljevic1992, Thm. pp.1] and colored Tverberg theorem of type B of Živaljević and Vrećica [Vrecica1994, Thm. 4]. In 2011 Blagojević, Matschke and Ziegler gave yet another colored topological transversal Tverberg theorem [Blagojevic2011-01, Thm. 1.3] that in the case coincides with their optimal colored Tverberg theorem [Blagojevic2009, Thm. 2.1].

The existence of counterexamples to the topological Tverberg conjecture for non-primepowers, obtained by Frick [Frick2015] [Blagojevic2015-02] based on the remarkable work of Mabillard and Wagner [Mabillard2014] [Mabillard2015], in particular invalidates Conjecture 1.2 in the case when and is not a prime power.

Acknowledgements.

We are grateful to Florian Frick and to the referee for very good observations and useful comments.

2. Statement of the main results

In 2014 Blagojević, Frick and Ziegler [Blagojevic2014] introduced the “constraint method,” by which the topological Tverberg theorem implies almost all its extensions, which had previously been obtained as substantial independent results, such as the “Colored Tverberg Theorem” of Živaljević and Vrećica [Zivaljevic1992] and the “Generalized Van Kampen–Flores Theorem” of Sarkaria [Sarkaria1991-1] and Volovikov [Volovikov1996-2]. Thus the constraint method reproduced basically all Tverberg type theorems obtained during more than three decades with a single elementary idea. Moreover, the constraint method in combination with the work of Mabillard and Wagner on the “-fold Whitney trick” [Mabillard2014] [Mabillard2015] yields counterexamples to the topological Tverberg theorem for non-prime powers, as demonstrated by Frick [Frick2015] [Blagojevic2015-02].

In this paper we use the constraint method in combination with a generalized Borsuk–Ulam type theorem and a cohomological intersection lemma to show how one can obtain many new topological transversal theorems of Tverberg type. We prove in detail a new generalized transversal van Kampen–Flores theorem and a new topological transversal weak colored Tverberg theorem.

Theorem 2.1 (The topological generalized transversal Van Kampen–Flores theorem).

Let

  • and be integers with ,

  • be powers of the prime , where are integers,

  • ,

  • , and

  • be even, or .

Then for every collection of continuous maps there exist an -dimensional affine subspace in and pairwise disjoint faces in the -skeleton of , for , such that

The special case of the previous theorem is due to Karasev [Karasev2007, Cor. 4].

In order to state the next result we recall the notion of a rainbow face. Suppose that the vertices of the simplex are partitioned into color classes . The subcomplex is called the rainbow complex, that is, the subcomplex of all faces that have at most one vertex of each color class . Faces of are called rainbow faces.

Theorem 2.2 (The topological transversal weak colored Tverberg theorem).

Let

  • and be integers with ,

  • be powers of the prime , where are integers,

  • ,

  • the vertices of the simplex , for every , be colored by colors, where each color class has cardinality at most ,

  • be even, or .

Then for every collection of continuous maps there exist an -dimensional affine subspace of and pairwise disjoint rainbow faces in , for , such that

The proofs of Theorems 2.1 and 2.2 are almost identical; see Sections LABEL:subsec_:_GVKF and LABEL:subsec_:_WCT. The only difference occurs in the definition of the bundles and , in (LABEL:def_of_xi_-_01) and (LABEL:def_of_xi_-_02), and the bundle maps for ; see Sections LABEL:subsubsec:_definition_of_function_GVKF and LABEL:subsubsec:_definition_of_function_WCT. Using the same proof technique as for these theorems and modifying the bundles and bundle maps using recipes from [Blagojevic2014, Lem. 4.2], one can also derive, for example, a topological transversal colored Tverberg theorem of type B, a topological transversal Tverberg theorem with equal barycentric coordinates, or mixtures of those. The most general transversal Tverberg theorem that is produced by the constraint method can be formulated using the concept of “Tverberg unavoidable subcomplexes” [Blagojevic2014, Def. 4.1], as follows.

Let , and be integers, and let be a continuous map with at least one Tverberg -partition, that is, a collection of pairwise disjoint faces such that . A subcomplex of the simplex is Tverberg unavoidable with respect to if for every Tverberg partition of there exists at least one face that lies in the subcomplex .

Theorem 2.3 (A constraint topological transversal Tverberg theorem).

Let

  • and be integers with ,

  • be integer,

  • be powers of the prime , where are integers,

  • ,

  • be a Tverberg unavoidable subcomplex of the simplex with respect to any continuous map for and , assuming that , and

  • be even, or .

Then for every collection of continuous maps there exist an -dimensional affine subspace of and pairwise disjoint faces that belong to the subcomplex , for , such that

3. A generalized Borsuk–Ulam type theorem and two lemmas

In this section, we present the topological methods, developed in [Blagojevic2011-01] and [Karasev2007], that we will use in the proofs of Theorems 2.1 and 2.2. In particular, we will review and slightly modify a generalized Borsuk–Ulam type theorem [Blagojevic2011-01, Thm. 4.1], give an intersection lemma [Blagojevic2011-01, Lem. 4.3] and recall the Euler class computation of Dol’nikov [Dolnikov1994, Lem. p. 2], Živaljević [Zivaljevic1999, Prop. 4.9], and Karasev [Karasev2007, Lem. 8].

3.1. Fadell–Husseini index

In 1988 Fadell and Husseini [Fadell1988] introduced an ideal-valued index theory for the category of -space, or more general for the category of -equivariant maps to a fixed space with a trivial -action. We give an overview of the index theory adjusted to the needs of this paper.

Let be a finite group, let be a commutative ring with unit, and let be a space with the trivial action. For a -equivariant map and a ring , the Fadell–Husseini index of is defined to be the kernel ideal of the map in the equivariant Čech cohomology with coefficients in the the ring induced by :

The equivariant cohomology of a -space is assumed to be the Čech cohomology of the Borel construction associated to the space .

The basic properties of the index are:

  • Monotonicity: If and are -equivariant maps, and is a -equivariant map such that , then

  • Additivity: If is an excisive triple of -spaces and is a -equivariant map, then

  • General Borsuk–Ulam–Bourgin–Yang theorem: Let and be -equivariant maps, and let be a -equivariant map such that . If then

In the case when is a point and is a -equivariant map we simplify notation and write . With this, the next property of the index can be formulated as follows.

  • If is a -space and is the projection on the first factor, then

3.2. A generalized Borsuk–Ulam type theorem

The cohomology of the elementary abelian groups , where is a prime and is an integer, is given by

The following theorem and its proof is just a slight modification of [Blagojevic2011-01, Thm. 4.1].

Theorem 3.1 (Borsuk–Ulam type theorem).

Let

  • be an elementary abelian group where is a prime and ,

  • be a connected space with the trivial -action,

  • be a -equivariant vector bundle where all fibers carry the same -representation,

  • be the fixed-point subbundle of the vector bundle ,

  • be its -invariant orthogonal complement subbundle ,

  • be the fiber of the vector bundle over the point ,

  • be the Euler class of the vector bundle , and

  • be a -CW-complex such that .

Assume that

  • acts trivially on , and

  • we are given a -equivariant map such that the following diagram commutes

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