Short proofs of Tverberg-type theorems for cell complexes

TL;DR

This paper provides concise proofs of Tverberg-type theorems applicable to cell complexes, extending classical results to more general topological spaces and simplifying the proof process.

Contribution

It introduces short, elegant proofs for Tverberg-type theorems on cell complexes, broadening the scope beyond simplicial complexes.

Findings

Valid for prime power r

Applicable to complexes homeomorphic to high-dimensional spheres

Guarantees intersection of images of disjoint faces under continuous maps

Abstract

We present short proofs of Tverberg-type theorems for cell complexes by S. Hasui, D. Kishimoto, M. Takeda, and M. Tsutaya. One of them states that for any prime power $r$, any complex $X$ topologically homeomorphic to $S^{(d+1)(r-1)-1}$, and any continuous map $f:X\to\mathbb R^d$ there are pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$ of $X$ such that $f(\sigma_1)\cap\ldots f(\sigma_r)\ne\emptyset$.