A tight colored Tverberg theorem for maps to manifolds (extended abstract)

Pavle V M Blagojević,Günter M Ziegler,Benjamin Matschke

doi:10.46298/dmtcs.2901

Abstract

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 $M$, assuming 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 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 $M=ℝ^d$. 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 proofs, as well as ours, work when r is a prime power. Étant donné un simplex $Δ _N$ de dimension $N$ ayant les sommets colorés, une face de $Δ _N$ est dite arc-en-ciel, si tous les sommets de cette face ont des couleurs différentes. Toute fonction continue d'un simplex $Δ _N$ de dimension $N$ aux sommets colorés vers une variété $d$-dimensionnelle $M$ doit envoyer $r$ points provenant de faces arc-en-ciel disjointes de $Δ _N$ au mêmes points dans $M$ ; en supposant que $N ≥(r-1)(d +1)$, un ensemble de $r$ sommets de $Δ _N$ doit être coloré à l'aide d'au moins deux couleurs. Notre démonstration requiert que $r$ soit un nombre premier. Ce résultat est une extension de notre "nouveau théorème de Tverberg coloré'', le cas particulier où $M = ℝ^d$. Il est également une généralisation du théorème de Tverberg topologique de Volovikov datant de 1996, pour les fonctions vers une variété, dont les classes de couleurs sont de taille 1 (c'est-à-dire sans contraintes de couleur). Dans ce cas particulier, la démonstration de Volovikov et la nôtre fonctionnent lorsque r est une puissance d'un premier.

Highlights

Theorem 1.1 is tight in the sense that it fails for maps of a simplex of smaller dimension, or if r vertices have the same color
The special case where all vertices of ∆N have different colors, |Ci| = 1, is the prime case of the topological Tverberg theorem, as proved by Barany, Shlosman & Szucs [4]. In this talk we present an extension of Theorem 1.1 that treats continuous maps R → M from the a subcomplex R of the N -simplex to an arbitrary d-dimensional manifold M with boundary in place of Rd
Since f maps the front face ∆N to M × {0} and since ∆N has only r − 1 < r vertices more than ∆N, already the Fi will intersect in M × {0}

Summary

Introduction

More than 50 years ago, the Cambridge undergraduate Bryan Birch [5] showed that “3N points in a plane” can be split into N triples that span triangles with a non-empty intersection. Theorem 1.1 is tight in the sense that it fails for maps of a simplex of smaller dimension, or if r vertices have the same color It implies an optimal result for the Barany–Larman conjecture in the case where r +1 is a prime, and an asymptotically-optimal bound in general; see [7, Corollaries 2.4, 2.5]. The special case where all vertices of ∆N have different colors, |Ci| = 1, is the prime case of the topological Tverberg theorem, as proved by Barany, Shlosman & Szucs [4] In this talk we present an extension of Theorem 1.1 that treats continuous maps R → M from the a subcomplex R of the N -simplex to an arbitrary d-dimensional manifold M with boundary in place of Rd. Here, R is the rainbow subcomplex ∆N , which consists of all rainbow faces. In the second step we rely on the computation of the Fadell–Husseini index of joins of chessboard complexes that we obtained in [8, Corollary 2.6]

A geometric reduction lemma

The Fadell–Husseini index

The index of the deleted join of the manifold