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Uspekhi Mat. Nauk, 2018, Volume 73, Issue 2(440), Pages 141–174 (Mi umn9774)  

A user's guide to the topological Tverberg conjecture

A. B. Skopenkovab

a Moscow Institute of Physics and Technology (State University)
b Independent University of Moscow

Abstract: 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 $\sigma_1,…,\sigma_r\subset\Delta$ such that $f(\sigma_1)\cap…\cap f(\sigma_r)\ne\varnothing$. 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.
Bibliography: 80 titles.

Keywords: multiple intersections, Tverberg theorem, Radon theorem, van Kampen–Flores theorem, Borsuk–Ulam theorem, configuration space, cohomology, equivariant maps, Whitney trick.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-06302
Dynasty Foundation
Simons Foundation
This research was supported by the Russian Foundation for Basic Research (grant no. 15-01-06302), the Simons-IUM Fellowship, and D. Zimin's Dynasty Foundation.
Subsection 3.2 was written jointly with R. Karasev. I am grateful to S. Avvakumov, P. Blagojević, V. Buchstaber, G. Kalai, R. Karasev, I. Mabillard, S. Melikhov, A. Ryabichev, M. Tancer, T. Tolozova and U. Wagner for useful remarks, and to I. Mabillard, and U. Wagner for allowing me to use some figures.


DOI: https://doi.org/10.4213/rm9774

Full text: PDF file (966 kB)
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English version:
Russian Mathematical Surveys, 2018, 73:2, 323–353

Bibliographic databases:

Document Type: Article
UDC: 515.143+519.178+514.174.5
MSC: 52A35, 05B99, 55S15, 55S35, 57Q35
Received: 24.03.2017
Revised: 01.02.2018

Citation: A. B. Skopenkov, “A user's guide to the topological Tverberg conjecture”, Uspekhi Mat. Nauk, 73:2(440) (2018), 141–174; Russian Math. Surveys, 73:2 (2018), 323–353

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