2023/02/28 by Feng, Ziqin, Nukala, Naga Chandra Padmini
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2302.14664
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, Fmn, Fnm∪ Fmn+1, Fnm∪ Fmn+2, and F\preceq Am. Here Fmn is the collection of size n subsets of [m] and F\preceq Am is the collection of subsets \preceq A where \preceq is a total order on the collections of subsets of [m] and A⊆ [m] (see the definition of \preceq in Section~\refIntro). We prove that the Vietoris-Rips complexes VR(Fmn, 2) and VR(Fnm∪ Fmn+1, 2) are either contractible or homotopy equivalent to a wedge sum of S2's; also, the complexes VR(Fnm∪ Fmn+2, 2) and VR(F\preceq Am, 2) are either contractible or homotopy equivalent to a wedge sum of S3's. We provide inductive formula for these homotopy types extending the result of Barmak in \citeBar13 about the independence complexes of Kneser graphs KG2, k and the result of Adamaszek and Adams in \citeAA22 about Vietoris-Rips complexes of hypercube graphs with scale 2.