A tight colored Tverberg theorem for maps to manifolds

TL;DR

This paper proves a new topological Tverberg theorem for maps from colored simplices to manifolds, establishing conditions under which multiple disjoint rainbow faces map to the same point, extending previous results to more general settings.

Contribution

It introduces a tight colored Tverberg theorem for maps to manifolds, generalizing earlier theorems and requiring the prime condition for r.

Findings

Proves that r points from disjoint rainbow faces map to the same point in a manifold.

Extends the colored Tverberg theorem to manifolds with specific dimensional conditions.

Requires r to be a prime for the theorem to hold.

Abstract

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.