Stronger counterexamples to the topological Tverberg conjecture
(1908.08731)Abstract
Denote by $\DeltaM$ the $M$-dimensional simplex. A map $f\colon \DeltaM\to\mathbb Rd$ is an almost $r$-embedding if $f\sigma1\cap\ldots\cap f\sigmar=\emptyset$ whenever $\sigma1,\ldots,\sigmar$ 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 $\Delta{(d+1)(r-1)}\to\mathbb Rd$. 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 $N:=(d+1)r-r\Big\lceil\dfrac{d+2}{r+1}\Big\rceil-2$, then there is an almost $r$-embedding $\DeltaN\to\mathbb Rd$. 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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