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A tight colored Tverberg theorem for maps to manifolds

Pavle V. M. Blagojević and Benjamin Matschke and Günter M. Ziegler – 2011

We prove that any continuous map of an N-dimensional simplex Delta_N with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of Delta_N to the same point in M: For this we have to assume that N \geq (r-1)(d+1), no r vertices of Delta_N get the same color, and our proof needs that r is a prime. A face of Delta_N 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=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 proof, as well as ours, work when r is a prime power.

Titel
A tight colored Tverberg theorem for maps to manifolds
Verfasser
Pavle V. M. Blagojević and Benjamin Matschke and Günter M. Ziegler
Datum
2011
Erschienen in
Topology and its Applications (Proc.\ ATA2010), volume 158, pages 1445-1452
Art
Text