Exploring homological properties of independence complexes of Kneser graphs

We discuss the topological properties of the independence complex of Kneser graphs, $Ind(KG(n,k))$, with n ≥ 3 and k ≥ 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 · 2n+k2n. Denote Fₙ[m] to be the collection of n-subsets of $[m]$ equipped with the symmetric difference metric. We prove that if l is the minimal integer with the qth dimensional reduced homology H̃q(VR(Fₙ[l];2(n-1))) being non-trivial, then rank(H̃q(VR(Fₙ[m];2(n-1))) ≥ ∑ᵢ₌ₗᵐi-2l-2 · rank(H̃q(VR(Fₙ[l];2(n-1))). Since we know the independence complex $Ind(KG(n,k))$ and the Vietoris-Rips complex VR(Fₙ[2n+k];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.