Stronger Counterexamples to the Topological Tverberg Conjecture
Stronger Counterexamples to the Topological Tverberg Conjecture
复制标题
拓扑特维尔伯格猜想的更强反例
作者：
S. Avvakumov;R. Karasev;A. Skopenkov
用 $$\Delta _M$$ Δ M 表示 M 维单纯形。映射 $$f:\Delta _M\rightarrow {{\mathbb {R}}}^d$$ f : Δ M → R d 几乎是 r 嵌入，如果 $$f(\sigma _1)\cap \ldots \cap f(\sigma _r)=\emptyset $$ f ( σ 1 ) ∩ … ∩ f ( σ r ) = ∅ 每当$$\sigma _1,\ldots ,\sigma _r$$ σ 1 , … , σ r 是成对不相交的面。拓扑 Tverberg 猜想的反例断言，如果 r 不是素数幂且 $$d\ge 2r+1$$ d ≥ 2 r + 1 ，则存在几乎 r 嵌入 $$\Delta _{(d+1)(r-1)}\rightarrow {{\mathbb {R}}}^d$$ Δ ( d + 1 ) ( r - 1 ) → R d 。 Blagojević-Frick-Ziegler 通过采用低维反例的 k 倍连接幂，使用高维反例的简单构造对此进行了改进。我们进一步改进这一点（对于与 r 相比较大的 d ）：如果 r 不是素数幂且 $$N=(d+1)r-r\Big \lceil \dfrac{d+2}{r+1}\Big \rceil -2$$ N = ( d + 1 ) r - r ⌈ d + 2 r + 1 ⌉ - 2 ，则存在几乎 r - 嵌入 $$\Delta _N\rightarrow {{\mathbb {R}}}^d$$ Δ N → R d 。这一改进源于我们对 r 折 van Kampen-Flores 猜想的更强反例。我们的证明基于关于从等变映射构建几乎 r 嵌入的马比拉德-瓦格纳定理以及关于等变映射存在的 Özaydin 定理的推广。
Denote by $$\Delta _M$$ Δ M the M -dimensional simplex. A map $$f:\Delta _M\rightarrow {{\mathbb {R}}}^d$$ f : Δ M → R d is an almost r - embedding if $$f(\sigma _1)\cap \ldots \cap f(\sigma _r)=\emptyset $$ f ( σ 1 ) ∩ … ∩ f ( σ r ) = ∅ whenever $$\sigma _1,\ldots ,\sigma _r$$ 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 $$d\ge 2r+1$$ d ≥ 2 r + 1 , then there is an almost r - embedding $$\Delta _{(d+1)(r-1)}\rightarrow {{\mathbb {R}}}^d$$ Δ ( d + 1 ) ( r - 1 ) → 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 $$N=(d+1)r-r\Big \lceil \dfrac{d+2}{r+1}\Big \rceil -2$$ N = ( d + 1 ) r - r ⌈ d + 2 r + 1 ⌉ - 2 , then there is an almost r - embedding $$\Delta _N\rightarrow {{\mathbb {R}}}^d$$ Δ N → R d . The improvement follows from our stronger counterexamples to the r -fold van Kampen–Flores conjecture. 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.
特韦尔贝格加上约束
影响因子：
0.9
作者：
Blagojević;Pavle V M;Florian;Ziegler;Günter M
通讯作者：
Günter M