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 , ... , m } equipped with symmetric difference metric d, specifically, F n m , F n m ∪ F n + 1 m , F n m ∪ F n + 2 m , and F ⪯ A m . Here F n m is the collection of size n subsets of [m] and F ⪯ A m is the collection of subsets ⪯ A where ⪯ is a total order on the collections of subsets of [m] and A ⊆ [ m ] (see the definition of ⪯ in Sect. 1). We prove that the Vietoris–Rips complexes V R (F n m , 2) and V R (F n m ∪ F n + 1 m , 2) are either contractible or homotopy equivalent to a wedge sum of S 2 's; also, the complexes V R (F n m ∪ F n + 2 m , 2) and V R (F ⪯ 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. [ABSTRACT FROM AUTHOR]