Čech complexes of hypercube graphs

Henry Adams ; Samir Shukla ; Anurag Singh

Homology, homotopy, and applications, Tome 27 (2025) no. 1, pp. 83-105.

Résumé

A Čech complex of a finite simple graph $G$ is a nerve complex of balls in the graph, with one ball centered at each vertex. More precisely, the Cech complex $\mathcal{N}(G, r)$ is the nerve of all closed balls of radius $\frac{r}{2}$ centered at vertices of $G$, where these balls are drawn in the geometric realization of the graph $G$ (equipped with the shortest path metric). The simplicial complex $\mathcal{N}(G, r)$ is equal to the graph $G$ when $r=1$, and homotopy equivalent to the graph $G$ when $r$ is smaller than half the length of the shortest cycle in $G$. For higher values of $r$, the topology of $\mathcal{N}(G, r)$ is not well-understood. We consider the $n$-dimensional hypercube graphs $\mathbb{I}_n$ with $2^n$ vertices. Our main results are as follows. First, when $r=2$, we show that the Čech complex $\mathcal{N}\left(\mathbb{I}_n, 2\right)$ is homotopy equivalent to a wedge of 2 spheres for all $n \geqslant 1$, and we count the number of 2 -spheres appearing in this wedge sum. Second, when $r=3$, we show that $\mathcal{N}\left(\mathbb{I}_n, 3\right)$ is homotopy equivalent to a simplicial complex of dimension at most 4 , and that for $n \geqslant 4$ the reduced homology of $\mathcal{N}\left(\mathbb{I}_n, 3\right)$ is nonzero in dimensions 3 and 4 , and zero in all other dimensions. Finally, we show that for all $n \geqslant 1$ and $r \geqslant 0$, the inclusion $\mathcal{N}\left(\mathbb{I}_n, r\right) \hookrightarrow \mathcal{N}\left(\mathbb{I}_n, r+2\right)$ is null-homotopic, providing a bound on the length of bars in the persistent homology of Čech complexes of hypercube graphs.

Détail

DOI : 10.4310/HHA.2025.v27.n1.a6

Classification : |55N31, 55U10, 05E45
Keywords: Čech complex, persistent homology, collapsibility, hypercube