Beyond the Borsuk–Ulam Theorem: The Topological Tverberg Story

Abstract

Bárány's "topological Tverberg conjecture" from 1976 states that any continuous map of an N -simplex ∆ N to R d , for N ≥ (d + 1)(r − 1), maps points from r disjoint faces in ∆ N to the same point in R d . The proof of this result for the case when r is a prime, as well as some colored version of the same result, using the results of Borsuk-Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matoušek's 2003 book "Using the Borsuk-Ulam theorem."In this paper we show how advanced equivariant topology methods allow one to go beyond the prime case of the topological Tverberg conjecture.First we explain in detail how equivariant cohomology tools (employing the Borel construction, comparison of Serre spectral sequences, Fadell-Husseini index, etc.) can be used to prove the topological Tverberg conjecture whenever r is a prime power. Our presentation includes a number of improved proofs as well as new results, such as a complete determination of the Fadell-Husseini index of chessboard complexes in the prime case.Then we introduce the "constraint method," which applied to suitable "unavoidable complexes" yields a great variety of variations and corollaries to the topological Tverberg theorem, such as the "colored" and the "dimension-restricted" (Van Kampen-Flores type) versions.Both parts have provided crucial components to the recent spectacular counter-examples in high dimensions for the case when r is not a prime power.

A user's guide to the topological Tverberg conjecture

“…The Constraint Lemma 3.2 (or, in Gromov's formulation, 'the topological Tverberg theorem, whenever available, implies the van Kampen-Flores theorem' [Gr10, 2.9.c, p. 445-446]) was proved in [Gr10,2.9.c,p.446,2nd paragraph] 4 and,independently,in [BFZ14,Lemma 4.1.iii and 4.2], [Fr,proof of Theorem 4]. In [BFZ14,BZ] the Constraint Lemma 3.2 is not explicitly stated but is implicitly proved in the proof of other results. Thus the lemma is proved separately for r a prime power [BFZ14,Lemma 4.1.iii and 4.2], [BZ,§4.1] or not [Fr,proof of Theorem 4], [BZ,§5], although neither case of the lemma uses the fact that r is a prime power or not.…”

“…In [BFZ14,BZ] the Constraint Lemma 3.2 is not explicitly stated but is implicitly proved in the proof of other results. Thus the lemma is proved separately for r a prime power [BFZ14,Lemma 4.1.iii and 4.2], [BZ,§4.1] or not [Fr,proof of Theorem 4], [BZ,§5], although neither case of the lemma uses the fact that r is a prime power or not.…”

“…In this subsection we list classical results motivating the topological Tverberg conjecture. For history, more motivation and related problems see [BBZ,Zi11], [BZ,[1][2][3], [Sk,§1.3 'Radon and Tverberg theorems']. This subsection is formally not used in the sequel.…”

General Bourgin–Yang theorems

“…We were also motivated by a recent exposition of various guises of the topological Tverberg theorem by Blagojević and Ziegler [3]. Its prime power version states that for any continuous map ∆ N → R m with N ≥ (m + 1)(p k − 1) there exist p k points in pairwise disjoint faces of the standard N-simplex ∆ N that are mapped to the same point of R m .…”

“…It is well-known that ∆ is an (N −p k +1)-dimensional (N −p k )-connected CW complex (see e.g. [3,Theorem 3.4]). Equip it with a G-action analogously as in Subsection 2.6 and consider a map f 1 : ∆ → R m given by f 1 (x 1 , .…”