Exploring Homological Properties of Independent Complexes of Kneser Graphs

We discuss the topological properties of the independence complex of Kneser graphs, Ind(KG$(n, k))$, with $n\geq 3$ and $k\geq 1$. By identifying one kind of maximal simplices through projective planes, we obtain homology generators for the $6$-dimensional homology of the complex Ind(KG$(3, k))$. Using cross-polytopal generators, we provide lower bounds for the rank of $p$-dimensional homology of the complex Ind(KG$(n, k))$ where $p=1/2\cdot {2n+k\choose 2n}$. Denote $\mathcal{F}_n^{[m]}$ to be the collection of $n$-subsets of $[m]$ equipped with the symmetric difference metric. We prove that if $\ell$ is the minimal integer with the $q$th dimensional reduced homology $\tilde{H}_q(\mathcal{VR}(\mathcal{F}^{[\ell]}_n; 2(n-1)))$ being non-trivial, then $$\text{rank} (\tilde{H}_q(\mathcal{VR}(\mathcal{F}_n^{[m]}; 2(n-1)))\geq \sum_{i=\ell}^m{i-2\choose \ell-2}\cdot \text{rank} (\tilde{H}_q(\mathcal{VR}(\mathcal{F}_n^{[\ell]}; 2(n-1))).$$ Since the independence complex Ind(KG$(n, k))$ and the Vietoris-Rips complex $\mathcal{VR}(\mathcal{F}^{[2n+k]}_n; 2(n-1))$ are the same, we obtain a homology propagation result in the setting of independence complexes of Kneser graphs. Connectivity of these complexes is also discussed in this paper.