Denote by \Delta_N the N-dimensional simplex. A map f\colon \Delta_N\to\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\ge2r+1, then there is an almost r-embedding (d+1)(r-1)\to\mathbb R^d.  We improve this by showing that if r is not a prime power and N:=(d+1)r-r\Big\lceild+2{r+1}\Big\rceil-2, then there is an almost r-embedding \Delta_N\to\mathbb R^d. For the r-fold van Kampen--Flores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the Mabillard--Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the \"Ozaydin theorem on existence of equivariant maps.
Denote by \Delta_N the N-dimensional simplex. A map f\colon \Delta_N\to\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\ge2r+1, then there is an almost r-embedding (d+1)(r-1)\to\mathbb R^d. This was improved by Blagojevi\'c-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\lceild+2{r+1}\Big\rceil-2, then there is an almost r-embedding \Delta_N\to\mathbb R^d. For the r-fold van Kampen-Flores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the Mabillard-Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the \"Ozaydin theorem on existence of equivariant maps.
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This was improved by Blagojevi\'c-Frick-Ziegler using a simple construction of higher-dimensional counterexamples by taking k-fold join power of lower-dimensional ones.
R
by showing that if r
further (for d large compared to r): If r
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Kampen--Flores
Kampen-Flores
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Mabillard--Wagner
Mabillard-Wagner