更强的拓扑Tverberg猜想反例
Stronger counterexamples to the topological Tverberg conjecture
AI总结：
本文提出更强的拓扑Tverberg猜想反例，通过改进方法构造更高维的反例，并基于Mabillard-Wagner定理和Ozaydin定理进行证明。
摘要：
令Δ_M表示M维单纯形。若σ₁,…,σ_r为 pairwise 不相交的面，则映射f:Δ_M→R^d称为几乎r-嵌入。反例于拓扑Tverberg猜想断言，若r不是素数幂且d≥2r+1，则存在一个几乎r-嵌入Δ_{(d+1)(r-1)}→R^d。Blagojević-Frick-Ziegler通过简单构造更高维反例的方法，将这一结论改进。我们进一步改进这一结果（当d远大于r时）：若r不是素数幂且N:=(d+1)r-r⌈(d+2)/(r+1)⌉-2，则存在一个几乎r-嵌入Δ_N→R^d。对于r重van Kampen-Flores猜想，我们也产生了比已知更强的反例。我们的证明基于Mabillard-Wagner定理关于从等价映射构造几乎r-嵌入的推广，以及Ozaydin定理关于等价映射存在的推广。
英文摘要：
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.
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