Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems

Abstract

Motivated by topological Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R d without triple, quadruple, or, more generally, r-fold points (image points with at least r distinct preimages), for a given multiplicity r ≥ 2. In particular, we are interested in maps f : K → R d that have no r-Tverberg

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].…”

Stronger counterexamples to the topological Tverberg conjecture

“…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.…”

“…Theorem 5 is a generalization of the Mabillard-Wagner theorem (see [MW15], [AMS+] and the survey [Sk16, Theorem 3.3]).…”

Topology of unavoidable complexes

“…Isaac Mabillard and Uli Wagner opened a new chapter of the theory of almost r-embeddings by introducing and developing a version of 'Whitney trick' for eliminating strong r-fold points. The following theorem was originally announced in [MW14] with the complete presentation given in [MW15], see also [AMSW,MW16] for the subsequent development.…”

“…Theorem 4.2. (I. Mabillard, U. Wagner [MW14,MW15]) Suppose that r ≥ 2, k ≥ 3, and let K be a simplicial complex of dimension (r − 1)k. Then the following statements are equivalent:…”