Vietoris-Rips and Čech Complexes of Certain Finite Metric Spaces

Abstract

We examine the homotopy types of Vietoris-Rips complexes on different collections of subsets of $[m]=\{1,2,\ldots,m\}$ equipped with the symmetric difference metric. More specifically, we prove that the Vietoris-Rips complexes $\V(\F;2)$ and $\V(\F\cup\FF;2)$ are either contractible or homotopy equivalent to a wedge sum of $S^2$'s. We also show that the complexes $\V(\F\cup \FFF;2)$ and $\V(\F_{\preceq A};2)$ are homotopy equivalent to a wedge sum of $S^3$'s. We found a more intuitive proof of the result of Adamamszek and Adams in \cite{AA22} about Vietoris-Rips complexes of hypercube graphs with scale $2$.

We also define \v{C}ech complexes on the finite union of finite metric spaces at scales $2,3$ equipped with symmetric difference metric. We prove that the \v{C}ech complexes\\ $\N(\F\cup \FF\cup \FFF;2)$ and $ \N(\F\cup \FF\cup \FFF;3)$ are either contractible or homotopy equivalent to a wedge sum of $S^2$'s and that the complex $\N(\F\cup \FF;2)$ is homotopy equivalent to a wedge sum of $S^1$'s. We also establish that the complex $\N(\f_0^m\cup \f_1^m\cup \f_2^m\cup \f_3^m;3)$ for $n=0$ is homotopy equivalent to a wedge sum of $S^4$'s. We provide (inductive) formulae for all these homotopy types.