## 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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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.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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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

### Keywords

- The topological Tverberg conjecture
- Multiple points of maps
- Equivariant maps
- Deleted product obstruction