Stronger counterexamples to the topological Tverberg conjecture
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arXiv
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| Format: | Preprint |
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2019
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| author | Avvakumov, S. Karasev, R. Skopenkov, A. |
| author_facet | Avvakumov, S. Karasev, R. Skopenkov, A. |
| contents | Denote by $Δ_M$ the $M$-dimensional simplex. A map $f\colon Δ_M\to\mathbb R^d$ is an almost $r$-embedding if $fσ_1\cap\ldots\cap fσ_r=\emptyset$ whenever $σ_1,\ldots,σ_r$ 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 $Δ_{(d+1)(r-1)}\to\mathbb R^d$. 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 $Δ_N\to\mathbb R^d$.
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 Özaydin theorem on existence of equivariant maps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1908_08731 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Stronger counterexamples to the topological Tverberg conjecture Avvakumov, S. Karasev, R. Skopenkov, A. Geometric Topology Computational Geometry Combinatorics 52C35, 55S91, 57S17 Denote by $Δ_M$ the $M$-dimensional simplex. A map $f\colon Δ_M\to\mathbb R^d$ is an almost $r$-embedding if $fσ_1\cap\ldots\cap fσ_r=\emptyset$ whenever $σ_1,\ldots,σ_r$ 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 $Δ_{(d+1)(r-1)}\to\mathbb R^d$. 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 $Δ_N\to\mathbb R^d$. 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 Özaydin theorem on existence of equivariant maps. |
| title | Stronger counterexamples to the topological Tverberg conjecture |
| topic | Geometric Topology Computational Geometry Combinatorics 52C35, 55S91, 57S17 |
| url | https://arxiv.org/abs/1908.08731 |