# Counterexamples to the topological Tverberg conjecture

@article{Frick2015CounterexamplesTT, title={Counterexamples to the topological Tverberg conjecture}, author={Florian Frick}, journal={arXiv: Combinatorics}, year={2015} }

The “topological Tverberg conjecture” by Barany, Shlosman and Sz˝ ucs (1981) states that any continuous map of a simplex of dimension (r 1)(d + 1) to R d maps points from r disjoint faces of the simplex to the same point in R d . This was established for affine maps by Tverberg (1966), for the case when r is a prime by Barany et al., and for prime power r by Ozaydin (1987). We combine the generalized van Kampen theorem announced by Mabillard and Wagner (2014) with the constraint method of…

## 58 Citations

Average-value tverberg partitions via finite fourier analysis

- Mathematics
- 2015

The long-standing topological Tverberg conjecture claimed, for any continuous map from the boundary of an N(q, d):= (q-1)(d+1)-simplex to d-dimensional Euclidian space, the existence of q pairwise…

Beyond the Borsuk–Ulam Theorem: The Topological Tverberg Story

- Mathematics
- 2017

Barany’s “topological Tverberg conjecture” from 1976 states that any continuous map of an N-simplex \(\Delta _{N}\) to \(\mathbb{R}^{d}\), for N ≥ (d + 1)(r − 1), maps points from r disjoint faces in…

A multiset extension of the optimal colored Tverberg theorem

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The type A colored Tverberg theorem of Blagojevic, Matschke, and Ziegler provides optimal bounds for the colored Tverberg problem, under the condition that the number of intersecting rainbow…

A user's guide to the topological Tverberg conjecture

- Mathematics
- 2018

A simplified explanation of easier parts of the arguments of the Tverberg conjecture is presented, accessible to non-specialists in the area, and reference to more complicated parts are given.

A Homotopy-Theoretic Approach to the Topological Tverberg Conjecture

- Mathematics
- 2017

Let r ≥ 2 and d ≥ 1 be integers, let N = (d + 1)(r − 1), and let ∆ denote a standard N -simplex. The Topological Tverberg Conjecture states that any continuous map f : ∆ → R has r-fold…

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

- MathematicsArXiv
- 2015

It is shown that under suitable restrictions on the dimensions, a well-known Deleted Product Criterion is not only necessary but also sufficient for the existence of maps without r-Tverberg points, which is a higher-multiplicity version of the classical Whitney trick.

A user's guide to disproof of topological Tverberg conjecture

- MathematicsArXiv
- 2016

A simplified explanation of easier parts of the counterexample of the Tverberg conjecture is presented, accessible to non-specialists in the area, and reference to more complicated parts are given.

The Crossing Tverberg Theorem

- MathematicsSoCG
- 2019

A strengthening of Tverberg's theorem is proved that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections.

Chromatic Numbers of Stable Kneser Hypergraphs via Topological Tverberg-Type Theorems

- MathematicsInternational Mathematics Research Notices
- 2018

Kneser’s 1955 conjecture—proven by Lovász in 1978—asserts that in any partition of the $k$-subsets of $\{1, 2, \dots , n\}$ into $n-2k+1$ parts, one part contains two disjoint sets. Schrijver…

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