论尺度为 2 的有限度量空间的 Vietoris-Rips 复数

Pub Date : 2024-02-03 DOI:10.1007/s40062-024-00340-x
Ziqin Feng, Naga Chandra Padmini Nukala
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引用次数: 0

摘要

我们研究了尺度为 2 的某些有限度量空间上的 Vietoris-Rips 复数的同调类型。我们考虑了配备对称差分度量 d 的 \([m]=\{1, 2, \ldots , m\}\) 子集的集合,特别是 \({\mathcal {F}}^m_n\)、\({\mathcal {F}}_n^m\up {\mathcal {F}}^m_{n+1}\),\({\mathcal {F}}_n^m\up {\mathcal {F}}^m_{n+2}\), and\({\mathcal {F}}_{p\receq A}^m\).这里,\({\mathcal {F}^m_n\) 是 [m] 的大小为 n 的子集的集合,\({\mathcal {F}_{\preceq A}^m\) 是子集的集合。其中 \(\preceq \)是[m]的子集集合的总序,而 \(A\subseteq [m]\)是[m]的子集集合(参见第 1 节中 \(\preceq \)的定义)。1).我们证明 Vietoris-Rips 复数 \({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}^m_n、2)\) 和 \({{{mathcal {V}}{{{mathcal {R}}}({\mathcal {F}_n^m\cup {mathcal {F}^m_{n+1}, 2)\) 要么是可收缩的,要么是等同于 \(S^2\) 的楔形和;此外,复数 ({{mathcal {V}}{{mathcal {R}}({\mathcal {F}}_n^m\cup {mathcal {F}}^m_{n+2}、2)\)和({{mathcal {V}}{{mathcal {R}}({\mathcal {F}_{\preceq A}^m, 2)\)要么是可收缩的,要么是与\(S^3\)的楔和等价的。我们提供了这些同调类型的归纳公式,扩展了巴马克关于 Kneser 图 KG\(_{2, k}\) 的独立性复数的结果,以及阿达马泽克和亚当斯关于尺度为 2 的超立方图的 Vietoris-Rips 复数的结果。
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On Vietoris–Rips complexes of finite metric spaces with scale 2

We examine the homotopy types of Vietoris–Rips complexes on certain finite metric spaces at scale 2. We consider the collections of subsets of \([m]=\{1, 2, \ldots , m\}\) equipped with symmetric difference metric d, specifically, \({\mathcal {F}}^m_n\), \({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+1}\), \({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+2}\), and \({\mathcal {F}}_{\preceq A}^m\). Here \({\mathcal {F}}^m_n\) is the collection of size n subsets of [m] and \({\mathcal {F}}_{\preceq A}^m\) is the collection of subsets \(\preceq A\) where \(\preceq \) is a total order on the collections of subsets of [m] and \(A\subseteq [m]\) (see the definition of \(\preceq \) in Sect. 1). We prove that the Vietoris–Rips complexes \({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}^m_n, 2)\) and \({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+1}, 2)\) are either contractible or homotopy equivalent to a wedge sum of \(S^2\)’s; also, the complexes \({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+2}, 2)\) and \({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}_{\preceq A}^m, 2)\) are either contractible or homotopy equivalent to a wedge sum of \(S^3\)’s. We provide inductive formulae for these homotopy types extending the result of Barmak about the independence complexes of Kneser graphs KG\(_{2, k}\) and the result of Adamaszek and Adams about Vietoris–Rips complexes of hypercube graphs with scale 2.

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