A tight colored Tverberg theorem for maps to manifolds |
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Authors: | Pavle V.M. Blagojevi?,Benjamin Matschke,Gü nter M. Ziegler |
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Affiliation: | a Mathemati?ki Institut, SANU, Knez Michailova 36, 11001 Beograd, Serbia b Institute of Mathematics, FU Berlin, Arnimallee 2, 14195 Berlin, Germany |
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Abstract: | We prove that any continuous map of an N-dimensional simplex ΔN with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of ΔN to the same point in M: For this we have to assume that N?(r−1)(d+1), no r vertices of ΔN get the same color, and our proof needs that r is a prime. A face of ΔN is 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=Rd. 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 proof, as well as ours, works when r is a prime power. |
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Keywords: | Colored Tverberg problem Deleted product/join configuration space Equivariant cohomology Fadell-Husseini index Serre spectral sequence |
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