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, . . . , m} equipped with symmetric difference metric d, specifically,Here F m n is the collection of size n subsets of [m] and F m A 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 Section 1). We prove that the Vietoris-Rips complexes VR(F m n , 2) and F m n ∪ F m n+1 are either contractible or homotopy equivalent to a wedge sum of S 2 's; also, the complexes VR(F m n ∪ F m n+2 , 2) and VR(F m A , 2) are either contractible or homotopy equivalent to a wedge sum of S 3 's. We provide inductive formula for these homotopy types extending the result of Barmak in [4] about the independence complexes of Kneser graphs KG 2,k and the result of Adamamszek and Adams in [2] about Vietoris-Rips complexes of hypercube graphs with scale 2.