A user's guide to the topological Tverberg conjecture

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 ${\sigma }_1,\dots ,{\sigma }_r\subset \Delta$ such that $f({\sigma }_1)\cap \dots \cap f({\sigma }_r)\ne \varnothing$. 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.

42 pages, 7 figures, exposition significantly improved and updated

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