A user's guide to the topological Tverberg conjecture

TL;DR

This paper provides an accessible overview of the topological Tverberg conjecture, its proof for prime powers, recent counterexamples for other values of r, and discusses related combinatorial and topological results.

Contribution

It offers a simplified, non-technical exposition of the conjecture, its proofs, counterexamples, and recent developments in the field.

Findings

The conjecture holds for prime power r.

Counterexamples exist for non-prime power r.

The interplay between combinatorics, algebra, and topology is crucial.

Abstract

The topological Tverberg conjecture was considered a central unsolved problem of topological combinatorics. The conjecture asserts that for any integers $r,d>1$ and any continuous map $f:\Delta\to\mathbb R^d$ of the $(d+1)(r-1)$-dimensional simplex there are pairwise disjoint faces $\sigma_1,\ldots,\sigma_r\subset\Delta$ such that $f(\sigma_1)\cap \ldots \cap f(\sigma_r)\ne\emptyset$. The conjecture was proved for a prime power $r$. Recently counterexamples for other $r$ were found. Analogously, the $r$-fold van Kampen-Flores conjecture holds for a prime power $r$ but does not hold for other $r$. The arguments form a beautiful and fruitful interplay between combinatorics, algebra and topology. We present a simplified exposition accessible to non-specialists in the area. We also mention some recent developments and open problems.