• Corpus ID: 119712823

Counterexamples to the topological Tverberg conjecture

@article{Frick2015CounterexamplesTT,
  title={Counterexamples to the topological Tverberg conjecture},
  author={Florian Frick},
  journal={arXiv: Combinatorics},
  year={2015}
}
  • F. Frick
  • Published 1 February 2015
  • Mathematics
  • arXiv: Combinatorics
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… 
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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.
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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.
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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
TLDR
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
  • F. Frick
  • Mathematics
    International 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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References

SHOWING 1-10 OF 13 REFERENCES
Eliminating Tverberg Points, I. An Analogue of the Whitney Trick
TLDR
This work considers multiple (double, triple, and higher multiplicity) self-intersection points of maps from finite simplicial complexes (compact polyhedra) into Rd, and study conditions under which such multiple points can be eliminated.
Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems
TLDR
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.
Barycenters of polytope skeleta and counterexamples to the Topological Tverberg Conjecture, via constraints
Using the authors' 2014 "constraints method," we give a short proof for a 2015 result of Dobbins on representations of a point in a polytope as the barycenter of points in a skeleton, and show that
Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry
A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not
A Generalization of Radon's Theorem
and proved it in the case n = 2. In the case n > 2, Birch proved a weaker result, with r(n+l) — n replaced by rn(n+l) — n — n+ 1. This was, for most r and n, an improvement of the earlier result, by
Tverberg plus constraints
Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this, we introduce a proof technique that
Equivariant Maps for the Symmetric Group
M. Ozaydin, Equivariant maps for the symmetric group, unpublished preprint, University of Wisconsin-Madison, 1987, 17
On the van Kampen-Flores theorem
In this paper we generalize the van Kampen-Flores theorem for mappings of a simplex into a topological manifold.
Ziegler , Tverberg plus constraints
  • Bull . Lond . Math . Soc .
  • 2014
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