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A tight colored Tverberg theorem for maps to manifolds (extended abstract)

Abstract : 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 $N≥(r-1)(d+1)$, 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.
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Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler. A tight colored Tverberg theorem for maps to manifolds (extended abstract). 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.183-190. ⟨hal-01215082⟩

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