多面体乘积的拓扑Tverberg定理

中文摘要 AI 辅助

拓扑Tverberg定理断言，若r为素数幂，则对于从$(r-1)(d+1)$维单纯形$\boldsymbol{\triangle_{(r-1)(d+1)}}$到$\boldsymbol{\reals^d}$的任意连续映射$\boldsymbol{f\function \triangle_{(r-1)(d+1)} \to \reals^d}$，存在该单纯形的r个两两不交的面，其像集具有非空的r重交集。通过细化可知，若将单纯形替换为同维数的任意多面体，上述结论依然成立。尽管该维数对单纯形而言是紧的，但Soberón与Zerbib的近期工作表明，这对一般多面体未必成立。本文中，我们给出单纯形乘积的拓扑Tverberg定理，其中每一定理均改进了$(r-1)(d+1)$维阈值，同时施加了“乘积的Tverberg面本身是各单纯形因子中两两不交面的乘积”这一结构条件。与前述情况类似，细化将这些结果（尤其是其维数改进）推广到任意多面体的乘积。作为示例，若$d+1$是2的幂，且$m\bgreq n\bgreq d+1$、$m+n=3d+2$，则我们证明任意连续映射$\boldsymbol{f\function \triangle_m \times \triangle_n \to \reals^d}$均存在$\triangle_m$的不交面$\boldsymbol{\tau_1,\tau_2}$与$\triangle_n$的不交面$\boldsymbol{\tau_1,\tau_2}$，使得$\boldsymbol{\bigcap_{i,j\bgin [2]} f(\tau_i \times \tau_j) \neq \nullset}$。对于多线性映射，我们的结果意味着$\reals^d$中网格索引点集的划分，其特殊子集的交集结论比Tverberg原定理更强。最后，我们将结果推广到对各乘积因子的面施加维数限制的van Kampen–Flores型定理。

英文摘要

The topological Tverberg theorem asserts that if $r$ is a prime power then for any continuous map $f\colon Δ_{(r-1)(d+1)}\rightarrow \mathbb{R}^d$ from the $(r-1)(d+1)$-dimensional simplex $Δ_{(r-1)(d+1)}$ to $\mathbb{R}^d$ there exist $r$ pairwise disjoint faces of the simplex whose images have non-empty $r$-fold intersection. By refinement, the same conclusion holds if the simplex is replaced by any polytope of the same dimension. While this dimension is tight for simplices, recent work of Soberón and Zerbib shows that this need not be true for polytopes in general. Here we give topological Tverberg theorems for products of simplices. Each of these improves upon the $(r-1)(d+1)$-dimensional threshold, even while imposing the structural condition that the ``Tverberg faces'' of the product are themselves the products of pairwise disjoint faces from each simplex factor. As before, refinement extends these results, and in particular their dimensional improvements, to products of arbitrary polytopes. As an example, if $d+1$ is a power of two then whenever $m\geq n\geq d+1$ and $m+n=3d+2$ we show that any continuous map $f\colon Δ_m\times Δ_n\rightarrow \mathbb{R}^d$ admits disjoint faces $σ_1,σ_2$ of $Δ_m$ and $τ_1,τ_2$ of $Δ_n$ such that $\cap_{i,j\in[2]} f(σ_i\timesτ_j)\neq \emptyset$. In the case of multilinear maps, our results imply partitions of grid-indexed point sets in $\mathbb{R}^d$ by specialized subsets with stronger intersection conclusions than given by Tverberg's original theorem. Lastly, we extend our results to van Kampen--Flores type theorems which impose dimensional restrictions on the faces of each product factor.

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20 pages