Pavle V. M. Blagojević ; Benjamin Matschke ; Günter M. Ziegler

A tight colored Tverberg theorem for maps to manifolds (extended abstract)
dmtcs:2901 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2011,
DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)

https://doi.org/10.46298/dmtcs.2901
A tight colored Tverberg theorem for maps to manifolds (extended abstract)
Authors: Pavle V. M. Blagojević ^{1}; Benjamin Matschke ^{2}; Günter M. Ziegler ^{2}
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Pavle V. M. Blagojević;Benjamin Matschke;Günter M. Ziegler
1 Mathematical Institute of the Serbian Academy of Sciences and Arts
2 Institut für Mathematik
Any continuous map of an $N$dimensional simplex $Δ _N$ with colored vertices to a $d$dimensional manifold $M$ must map $r$ points from disjoint rainbow faces of $Δ _N$ to the same point in $M$, assuming that $N≥(r1)(d+1)$, no $r$ vertices of $Δ _N$ get the same color, and our proof needs that $r$ is a prime. A face of $Δ _N$ is called a rainbow face if all vertices have different colors. This result is an extension of our recent "new colored Tverberg theorem'', the special case of $M=ℝ^d$. It is also a generalization of Volovikov's 1996 topological Tverberg theorem for maps to manifolds, which arises when all color classes have size 1 (i.e., without color constraints); for this special case Volovikov's proofs, as well as ours, work when r is a prime power.
Advanced Techniques of Cryptology, Image Processing and Computational Topology for Information Security; Funder: Ministry of Education, Science and Technological Development of Republic of Serbia; Code: 174008
Structured Discrete Models as a basis for studies in Geometry, Numerical Analysis, Topology, and Visualization; Funder: European Commission; Code: 247029