Counterexamples to the topological Tverberg conjecture

Abstract

The "topological Tverberg conjecture" by Bárány, Shlosman and Szűcs (1981) states that any continuous map of a simplex of dimension $(r-1)(d+1)$ to ${\mathbb{R}}^d$ maps points from $r$ disjoint faces of the simplex to the same point in ${\mathbb{R}}^d$. This was established for affine maps by Tverberg (1966), for the case when $r$ is a prime by Bárány et al., and for prime power $r$ by Özaydin (1987). We combine the generalized van Kampen theorem announced by Mabillard and Wagner (2014) with the constraint method of Blagojević, Ziegler and the author (2014), and thus prove the existence of counterexamples to the topological Tverberg conjecture for any number $r$ of faces that is not a prime power. However, these counterexamples require that the dimension $d$ of the codomain is sufficiently high: the smallest counterexample we obtain is for a map of the $100$-dimensional simplex to ${\mathbb{R}}^{19}$, for $r=6$.

3 pages, to appear in Oberwolfach Reports. This version differs from the Oberwolfach Reports version in two updated references. This manuscript is now part of arXiv:1510.07984

References in corpus (1)

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

Cited by in corpus (16)

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• The topological Tverberg problem beyond prime powers
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• Topology of unavoidable complexes
• Regular Polygonal Partitions of a Tverberg Type
• Tverberg's theorem for cell complexes
• On Reay's relaxed Tverberg conjecture and generalizations of Conway's thrackle conjecture
• Quantitative Tverberg theorems over lattices and other discrete sets
• Low stages of the Taylor tower for r-immersions
• Neighboring mapping points theorem
• Tverberg type theorems for matroids
• Quantitative combinatorial geometry for continuous parameters
• Topological Colored Tverberg Theorem and the Reduction Lemma
• Colored Tverberg problem, extensions and new results
• A `converse' to the Constraint Lemma