Topological Tverberg Theorem Research Articles

Tverberg's theorem for cell complexes

AbstractThe topological Tverberg theorem states that any continuous map of a ‐simplex into the Euclidean ‐space maps some points from pairwise disjoint faces of the simplex to the same point whenever is a prime power. We substantially generalize this theorem to continuous maps of certain CW complexes, including simplicial ‐spheres, into the Euclidean ‐space. We also discuss the atomicity of the Tverberg property.

Topological Tverberg Theorem: the proofs and the counterexamples

The history of the Topological Tverberg Theorem (TTT) is described. Some important constructions are presented, and their properties are discussed. In particular, there is a detailed description of the cell structure of the classifying space , where is the permutation group. Bibliography: 9 titles.

Optimal bounds for the colored Tverberg problem

We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of B´ar´any & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.

Tverberg plus constraints

Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this, we introduce a proof technique that combines a concept of ‘Tverberg unavoidable subcomplexes’ with the observation that Tverberg points that equalize the distance from such a subcomplex can be obtained from maps to an extended target space. Thus, we obtain simple proofs for many variants of the topological Tverberg theorem, such as the colored Tverberg theorem of Živaljević and Vrećica (‘The colored Tverberg's problem and complexes of injective functions’, J. Combin. Theory, Ser. A 61 (1992) 309–318). We also get a new strengthened version of the generalized van Kampen–Flores theorem by Sarkaria (‘A generalized van Kampen–Flores theorem’, Proc. Amer. Math. Soc. 11 (1991) 559–565) and Volovikov (‘On the van Kampen–Flores theorem’, Math. Notes ( 5 ) 59 (1996) 477–481), an affine version of their ‘ j-wise disjoint’ Tverberg theorem, and a topological version of Soberón's result (‘Equal coefficients and tolerance in coloured Tverberg partitions’, Proceeding of the 29th Annual Symposium on Computational Geometry (SoCG) (ACM, Rio de Janeiro, 2013), 91–96) on Tverberg points with equal barycentric coordinates.

Tverberg’s Theorem and Graph Coloring

The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in the same part. If the restrictions are too harsh, then the topological Tverberg theorem fails. The colored Tverberg theorem corresponds to graphs constructed as disjoint unions of small complete graphs. Hell studied the case of paths and cycles. In graph theory these partitions are usually viewed as graph colorings. As explored by Aharoni, Haxell, Meshulam and others there are fundamental connections between several notions of graph colorings and topological combinatorics. For ordinary graph colorings it is enough to require that the number of colors q satisfy q>Δ, where Δ is the maximal degree of the graph. It was proven by the first author using equivariant topology that if q>Δ2 then the topological Tverberg theorem still works. It is conjectured that q>KΔ is also enough for some constant K, and in this paper we prove a fixed-parameter version of that conjecture. The required topological connectivity results are proven with shellability, which also strengthens some previous partial results where the topological connectivity was proven with the nerve lemma.

The topological Tverberg theorem and related topics

The classical Borsuk–Ulam theorem, established some eighty years ago, may now be seen as a consequence of the nonvanishing of the mod 2 cohomology Euler class of a certain vector bundle over a real projective space. A theorem of Kakutani from the 1940s that any continuous real-valued function on the 2–sphere must be constant on some set of three orthogonal vectors may be deduced similarly from the nontriviality of some mod 3 cohomology Euler class. The more recent topological Tverberg theorem of Barany, Shlosman and Szucs, concerning a prime p, and the extensions of that theorem which have appeared in the last few years in the work of Blagojevic, Karasev, Matschke, Ziegler and others, may be proved by showing that some mod p Euler class is nonzero. This paper presents a survey of these, and related, results from the viewpoint of topological fibrewise fixed–point theory.

A tight colored Tverberg theorem for maps to manifolds

We prove that any continuous map of an N-dimensional simplex Δ N with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of Δ N to the same point in M: For this we have to assume that N ⩾ ( r − 1 ) ( d + 1 ) , no r vertices of Δ N get the same color, and our proof needs that r is a prime. A face of Δ N is a rainbow face if all vertices have different colors. This result is an extension of our recent “new colored Tverberg theorem”, the special case of M = R d . It is also a generalization of Volovikovʼs 1996 topological Tverberg theorem for maps to manifolds, which arises when all color classes have size 1 (i.e., without color constraints); for this special case Volovikovʼs proof, as well as ours, works when r is a prime power.

A local criterion for Tverberg graphs

The topological Tverberg theorem states that for any prime power q and continuous map from a (d+1)(q−1)-simplex to ℝd, there are q disjoint faces F i of the simplex whose images intersect. It is possible to put conditions on which pairs of vertices of the simplex that are allowed to be in the same face F i . A graph with the same vertex set as the simplex, and with two vertices adjacent if they should not be in the same F i , is called a Tverberg graph if the topological Tverberg theorem still work.These graphs have been studied by Hell, Schöneborn and Ziegler, and it is known that disjoint unions of small paths, cycles, and complete graphs are Tverberg graphs. We find many new examples by establishing a local criterion for a graph to be Tverberg. An easily stated corollary of our main theorem is that if the maximal degree of a graph is D, and D(D+1)<q, then it is a Tverberg graph.We state the affine versions of our results and also describe how they can be used to enumerate Tverberg partitions.

A tight colored Tverberg theorem for maps to manifolds (extended abstract)

Any continuous map of an $N$-dimensional simplex $Δ _N$ with colored vertices to a $d$-dimensional manifold $M$ must map $r$ points from disjoint rainbow faces of $Δ _N$ to the same point in $M$, assuming that $N≥(r-1)(d+1)$, no $r$ vertices of $Δ _N$ get the same color, and our proof needs that $r$ is a prime. A face of $Δ _N$ is called a rainbow face if all vertices have different colors. This result is an extension of our recent "new colored Tverberg theorem'', the special case of $M=ℝ^d$. It is also a generalization of Volovikov's 1996 topological Tverberg theorem for maps to manifolds, which arises when all color classes have size 1 (i.e., without color constraints); for this special case Volovikov's proofs, as well as ours, work when r is a prime power. Étant donné un simplex $Δ _N$ de dimension $N$ ayant les sommets colorés, une face de $Δ _N$ est dite arc-en-ciel, si tous les sommets de cette face ont des couleurs différentes. Toute fonction continue d'un simplex $Δ _N$ de dimension $N$ aux sommets colorés vers une variété $d$-dimensionnelle $M$ doit envoyer $r$ points provenant de faces arc-en-ciel disjointes de $Δ _N$ au mêmes points dans $M$ ; en supposant que $N ≥(r-1)(d +1)$, un ensemble de $r$ sommets de $Δ _N$ doit être coloré à l'aide d'au moins deux couleurs. Notre démonstration requiert que $r$ soit un nombre premier. Ce résultat est une extension de notre "nouveau théorème de Tverberg coloré'', le cas particulier où $M = ℝ^d$. Il est également une généralisation du théorème de Tverberg topologique de Volovikov datant de 1996, pour les fonctions vers une variété, dont les classes de couleurs sont de taille 1 (c'est-à-dire sans contraintes de couleur). Dans ce cas particulier, la démonstration de Volovikov et la nôtre fonctionnent lorsque r est une puissance d'un premier.

Tverberg's theorem with constraints

The topological Tverberg theorem claims that for any continuous map of the ( q − 1 ) ( d + 1 ) -simplex σ ( d + 1 ) ( q − 1 ) to R d there are q disjoint faces of σ ( d + 1 ) ( q − 1 ) such that their images have a non-empty intersection. This has been proved for affine maps, and if q is a prime power, but not in general. We extend the topological Tverberg theorem in the following way: Pairs of vertices are forced to end up in different faces. This leads to the concept of constraint graphs. In Tverberg's theorem with constraints, we come up with a list of constraints graphs for the topological Tverberg theorem. The proof is based on connectivity results of chessboard-type complexes. Moreover, Tverberg's theorem with constraints implies new lower bounds for the number of Tverberg partitions. As a consequence, we prove Sierksma's conjecture for d = 2 and q = 3 .

The Topological Tverberg Theorem and winding numbers

The Topological Tverberg Theorem claims that any continuous map of a ( q - 1 ) ( d + 1 ) -simplex to R d identifies points from q disjoint faces. (This has been proved for affine maps, for d ⩽ 1 , and if q is a prime power, but not yet in general.) The Topological Tverberg Theorem can be restricted to maps of the d-skeleton of the simplex. We further show that it is equivalent to a “Winding Number Conjecture” that concerns only maps of the ( d - 1 ) -skeleton of a ( q - 1 ) ( d + 1 ) -simplex to R d . “Many Tverberg partitions” arise if and only if there are “many q-winding partitions.” The d = 2 case of the Winding Number Conjecture is a problem about drawings of the complete graphs K 3 q - 2 in the plane. We investigate graphs that are minimal with respect to the winding number condition.

Notes on the topological Tverberg theorem

Following a manuscript of K.S. Sarkaria we give an extensive account of the topological Tverberg Theorem in the case where the number of desired disjoint faces is a prime power. Since all known proofs of the theorem utilize methods from algebraic topology, the objective of this article is to explain the proof to a wider mathematical audience.