math_groups_discgeom

A tight colored Tverberg theorem for maps to manifolds

Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler— 2011

Focus Area 3: Topological connectivity and diameter of Discrete Structures 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 Volovikovʼs 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 Volovikovʼs proof, as well as ours, works when r is a prime power.

TitleA tight colored Tverberg theorem for maps to manifolds
AuthorPavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler
Date20110801
Source(s)
Appeared InTopology and its Applications, Volume 158, Issue 12, 1 August 2011, Pages 1445–1452
TypeText