A tight colored Tverberg theorem for maps to manifolds (extended abstract)
Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler – 2011
Focus Area 3: Topological connectivity and diameter of Discrete Structures 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, assuming 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 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=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 proofs, as well as ours, work when r is a prime power.