Stronger counterexamples to the topological Tverberg conjecture
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Denote by Δ_N the N-dimensional simplex. A map fΔ_N→R^d is an almost r-embedding if fσ_1∩...∩ fσ_r=∅ whenever σ_1,...,σ_r are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and d>2r+1, then there is an almost r-embedding Δ_(d+1)(r-1)→R^d. We improve this by showing that if r is not a prime power and N:=(d+1)r-rd+2r+1-2, then there is an almost r-embedding Δ_N→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 Özaydin theorem on existence of equivariant maps.
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