A tight colored Tverberg theorem for maps to manifolds

Abstract

We prove that 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: For this we have to assume 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 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 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.