On Vietoris–Rips complexes of finite metric spaces with scale 2

Abstract

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.