Cohomological properties of the Vietoris--Rips Complex of a Hypercube Graph

TL;DR

This paper introduces a topological and combinatorial framework to analyze the cohomology of Vietoris--Rips complexes of hypercube graphs, providing new bounds and counterexamples to existing conjectures.

Contribution

It develops a novel toric topological approach, constructs explicit cohomology classes, and introduces ghost vertices to study Vietoris--Rips complexes of hypercubes.

Findings

Established lower bounds on connectivity of complexes.

Provided counterexamples to Shukla's conjecture.

Constructed explicit cohomology classes with combinatorial realizations.

Abstract

We develop a toric topological framework for studying the cohomology of Vietoris--Rips complexes $VR(Q_n;r)$ of hypercube graphs. Using total domination invariants and spectral methods, we establish general lower bounds on connectivity, which leads to infinite families of counterexamples to Shukla's conjecture, and derive first global upper bounds on coconnectivity. Our approach interprets Vietoris--Rips complexes via Stanley--Reisner rings, moment-angle complexes, and Tor algebras, allowing global topological information to be extracted from combinatorial data. In a second direction, we construct explicit cohomology classes using the Koszul resolution and show that they decomposable products of $1$-dimensional classes, and that their representatives can be combimbinatorially realised as the boundary of cross polytopes positively answering the question posed by Adams and Virk. We introduce ghost vertices as a new tool for detecting, extending, and proving linear independence of cohomology classes.