Symmetric multiple chessboard complexes and a new theorem of Tverberg type

TL;DR

This paper introduces a new Tverberg-type theorem confirming a conjecture about balanced partitions, utilizing properties of symmetric multiple chessboard complexes.

Contribution

It proves a conjecture on balanced Tverberg partitions using connectivity and shellability of symmetric multiple chessboard complexes.

Findings

Confirmed the conjecture on balanced Tverberg partitions

Established connectivity and shellability of symmetric multiple chessboard complexes

Provided new combinatorial topological tools for Tverberg-type theorems

Abstract

We prove a new theorem of Tverberg type which confirms the conjecture of Blagojevic, Frick, and Ziegler about the existence of "balanced Tverberg partitions" (Conjecture 6.6 in, Tverberg plus constraints, Bull. London Math. Soc., 46 (2014) 953-967). The proof relies on the connectivity and shellability properties of multiple chessboard complexes and their symmetric analogues.