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Stronger Counterexamples to the Topological Tverberg Conjecture

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Abstract

Denote by \(\Delta _M\) the M-dimensional simplex. A map \(f:\Delta _M\rightarrow {{\mathbb {R}}}^d\) is an almost r-embedding if \(f(\sigma _1)\cap \ldots \cap f(\sigma _r)=\emptyset \) whenever \(\sigma _1,\ldots ,\sigma _r\) are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and \(d\ge 2r+1\), then there is an almost r-embedding \(\Delta _{(d+1)(r-1)}\rightarrow {{\mathbb {R}}}^d\). This was improved by Blagojević–Frick–Ziegler using a simple construction of higher-dimensional counterexamples by taking k-fold join power of lower-dimensional ones. We improve this further (for d large compared to r): If r is not a prime power and \(N=(d+1)r-r\Big \lceil \dfrac{d+2}{r+1}\Big \rceil -2\), then there is an almost r-embedding \(\Delta _N\rightarrow {{\mathbb {R}}}^d\). The improvement follows from our stronger counterexamples to the r-fold van Kampen–Flores conjecture. Our proof is based on generalizations of the Mabillard–Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the Özaydin theorem on existence of equivariant maps.

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Notes

  1. For a finite cyclic or dihedral group G, and a certain representation space V of G, G-equivariant maps from the classifying space EG to \(V-0\) were constructed in [8]. Theorem 2.2 should also be compared to [4, Theorem 3.6 and the paragraph afterwards]. That reference takes a group G from a certain class and proves that there exists some representation W of G, for which there exist G-equivariant maps \(X \rightarrow S(W)\) for certain G-spaces X. However, \(G=\Sigma _r\) does not belong to that class, and the \(\Sigma _r\)-space S(W) described in [4, Theorem 3.6 and the paragraph afterwards] need not coincide with the \(\Sigma _r\)-space \({{\mathbb {R}}}^{2\times r}-\delta _r\) given by Theorem 2.2.

  2. E.g. take \(h_{-,t}(x) = \frac{h_{-,1/2}(x)}{2-2t+(2t-1)|h_{-,1/2}(x)|}\).

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Acknowledgements

We are grateful to M. Berezovik, F. Frick, A. Magazinov, and the anonymous referees for helpful suggestions.

Funding

S. Avvakumov: Supported by the Austrian Science Fund (FWF), Project P31312-N35 and the European Research Council under the European Union’s Seventh Framework Programme ERC Grant agreement ERC StG 716424–CASe.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George St., Toronto, ON, M5S 2E4, Canada

    S. Avvakumov

  2. Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia, 127994

    R. Karasev

  3. Moscow Institute of Physics and Technology, Institutskiy per., Dolgoprudnyi, Russia, 141700

    A. Skopenkov

  4. Independent University of Moscow, B. Vlasyevskiy, 11, Moscow, Russia, 119002

    A. Skopenkov

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  1. S. Avvakumov
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Correspondence to R. Karasev.

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Avvakumov, S., Karasev, R. & Skopenkov, A. Stronger Counterexamples to the Topological Tverberg Conjecture. Combinatorica 43, 717–727 (2023). https://doi.org/10.1007/s00493-023-00031-w

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  • DOI: https://doi.org/10.1007/s00493-023-00031-w

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