A user's guide to the topological Tverberg conjecture

Abstract

The well-known topological Tverberg conjecture was considered a central unsolved problem of topological combinatorics. The conjecture asserts that for each integers r, d > 1 and each continuous map f :A proof for a prime power r was given by I. Bárány, S. Shlosman, A. Szűcs, M. Özaydin and A. Volovikov in 1981-1996. A counterexample for other r was found in a series of papers most of them recent. (The exact description of contribution of particular authors is more complex and we provide more details in historical notes.)The arguments form a beautiful and fruitful interplay between combinatorics, algebra and topology. In this expository note we present a simplified explanation of easier parts of the arguments, accessible to non-specialists in the area.1 The exact description of contribution of particular authors is more complex and we provide more details in §3.2. We do not claim that the contributions were equal, but leave it to a reader to make his/her own opinion. M. de Longueville is not included in this list because Lemma 3.1 is used not to produce a counterexample, but to construct counterexamples for d > 3r + 1 from a counterexample for d = 3r + 1.

Colored Tverberg problem, extensions and new results

“…Tverberg-Van Kampen-Flores type results have been for decades one of the central research themes in topological combinatorics. The last decade has been particularly fruitful, bringing the resolution (in the negative) of the general "Topological Tverberg Problem" [MW14,F,BFZ2,MW15,MW16], as summarized by several review papers [BBZ,BS,Sk18,Ž17].…”

“…We refer the reader to [ŽV92,VŽ94,BMZ], [M03, Živ98] and [BBZ,BS,Sk18,Ž17] for more general statements, proofs, history and applications of colored Tverberg theorems.…”

Hardness of almost embedding simplicial complexes in $\mathbb R^d$

“…In this paper we study almost embeddings and equivariant maps of configuration spaces. They appear in studies of embeddings [FKT94,Sko08] as well as in topological combinatorics (for Tverberg-type problems see [BZ16,BBZ16,Sko16]). Almost embeddings also turned out to be a useful tool for studying Helly-type results on convex sets, implicitly in [Mat97] and explicitly in [GPP + 15].…”

Stronger counterexamples to the topological Tverberg conjecture

“…, σ r are pairwise disjoint faces of K. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and d ≥ 2r + 1, then there is an almost r-embedding ∆ (d+1)(r−1) → R d . See the surveys [BZ16,Sk16] and the references therein.…”

“…By general position, any k-complex admits an almost r-embedding in R k+ k+1 r−1 . A counterexample to the r-fold van Kampen-Flores conjecture asserts that if r is not a prime power and k is divisible by r − 1, then any k-complex admits an almost r-embedding in R k+ k r−1 (this is a combination of results of Özaydin [Oz] and Mabillard-Wagner [MW15], see the survey [Sk16]). Theorem 3 produces stronger counterexamples to the r-fold van Kampen-Flores conjecture.…”