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.

我们在尺度 2 的某些有限度量空间上检查 Vietoris-Rips 复形的同伦类型。我们考虑配备对称差的\([m]=\{1, 2, \ldots , m\}\)子集的集合度量d，具体来说，\({\mathcal {F}}^m_n\)、\({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+1}\)、\({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+2}\)和\({\mathcal {F}}_{\preceq A}^m\ ）。这里\({\mathcal {F}}^m_n\)是[ m ] 大小为n个子集的集合，而\({\mathcal {F}}_{\preceq A}^m\)是子集的集合\(\preceq A\)其中\(\preceq \)是 [ m ] 和\(A\subseteq [m]\)的子集集合上的全序（参见\(\preceq \)的定义第 1 节）。我们证明 Vietoris-Rips 复形\({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}^m_n, 2)\)和\({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+1}, 2)\) 是可收缩的或同伦等价于\(S^2\)的楔形和；另外，复数\({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+2 }, 2)\)和\({{\mathcal {V}}}{{\mathcal {R}}}({\mathcal {F}}_{\preceq A}^m, 2)\)是可收缩或同伦等价于\(S^3\)的楔和。我们提供了这些同伦类型的归纳公式，扩展了 Barmak 关于 Kneser 图 KG \(_{2, k}\)的独立复形的结果以及 Adamaszek 和 Adams 关于尺度为 2 的超立方图的 Vietoris-Rips 复形的结果。