DMS Topology and Geometry Seminar

Speaker: Prof. Ziqin Feng (Auburn University)

Title: Homotopy Types of Vietoris–Rips Complexes from Some Structured Graphs

Abstract: We investigate the topological properties of Vietoris–Rips complexes of the hypercube graph \(Q_n\)  equipped with the shortest-path metric. The hypercube’s highly symmetric structure make it a natural setting for studying the interaction between combinatorial geometry and topological complexity in Vietoris–Rips filtrations. We first analyze the homotopy types of these complexes at small scale parameters \(r=1, 2, 3\), providing explicit descriptions of their homotopy types.

For larger scale parameters, we relate the appearance of higher-dimensional homology to classical constructions in combinatorial design theory. In particular, we show that the existence of Hadamard matrices governs the emergence of nontrivial homology in high dimensions. More precisely, we prove that the \((2r−1)\)^th homology group of the Vietoris–Rips complex \(VR(Q_n;r)\) nontrivial for all sufficiently large \(n\), provided that a Hadamard matrix of order \(2r\) exists. This connection reveals a surprising link between the topology of Vietoris–Rips complexes on hypercube graphs and the existence of highly structured combinatorial objects.