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.