A user's guide to the topological Tverberg conjecture

The well-known topological Tverberg conjecture was considered a central unsolved problem of topological combinatorics. The conjecture asserts that for any integers $r$, $d$ and any continuous map $f\colon\Delta\to\mathbb{R}^d$ 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 $r$, but recently counterexamples for other $r$ were found. Similarly, the $r$-fold van Kampen–Flores conjecture holds for a prime power $r$ but not for other $r$. The arguments form a beautiful and fruitful interplay among combinatorics, algebra, and topology. This survey presents a simplified exposition accessible to non-specialists in the area, along with some recent developments and open problems.
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