Lower Bounds for Betti Numbers in Vietoris–Rips Complexes of Hypercubes

Abstract

This thesis serves as a comprehensive introduction and elucidation of Henry Adams and Žiga Virk's seminal work \cite{Lower bounds} on new lower bounds on the Betti numbers for Vietoris–Rips complexes of hypercube graphs across all dimensions and scales. Specifically, for a hypercube graph of dimension $n$ with vertex set $Q_n$ comprising $2^n$ vertices and equipped with the shortest path metric, we examine its Vietoris–Rips complex $\operatorname{VR}(Q_n;r)$ at any given scale parameter $r \geq 0$. Here, $\operatorname{VR}(Q_n;r)$ includes $Q_n$ as its vertex set and considers all subsets with a maximum diameter of $r$ as its simplices. Given integers $r < r'$, the inclusion $\operatorname{VR}(Q_n;r) \hookrightarrow \operatorname{VR}(Q_n;r')$ is found to be nullhomotopic, indicating that persistent homology bars do not extend beyond a unit length. Consequently, the study concentrates on the individual spaces $\operatorname{VR}(Q_n;r)$. By succinctly presenting the foundational definitions and correcting minor inaccuracies in their formulation, we aim to make Adams and Virk's work more accessible and understandable. And we introduce Adams and Virk's work on lower bounds for the ranks of a specific dimensional homology group on these complexes. Utilizing cross-polytopal generators, for instance, we ascertain that the rank of $H_{2^r-1}(\operatorname{VR}(Q_n;r))$ is no less than $2^{n-(r+1)} \binom{n}{r+1}$.