Homotopy types of Vietoris-Rips complexes of Hypercube Graphs

Abstract: We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at scale $3$. We represent the vertices in the hypercube graph $Q_m$ as the collection of all subsets of $[m]={1, 2, \ldots, m}$ and equip $Q_m$ with the metric using symmetric difference distance. It is proved in \cite{AA22} that the Vietoris-Rips complexes of hypercube graphs $Q_m$ at scale $2$, $\mathcal{VR}(Q_m; 2)$, is homotopy equivalent to $c_m$-many spheres with dimension $3$ where $c_m=\sum_{0\leq j< i<m} (j+1)(2^{m-2}-2^{i-1})$. Questions are raised in \cite{AA22} for determining the homotopy types of $\mathcal{VR}(Q_m, r)$ with large scales $r=3, 4, \ldots, m-2$. We prove that for $m\geq 5$, $$\mathcal{VR}(Q_m; 3)\simeq (\bigvee_{2^{m-4}\cdot{m\choose 4}} S^7) \vee (\bigvee_{\sum_{i=4}^{m-1}2^{i-4}\cdot{i\choose 4}} S^4).$$