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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+k 2n. Denote Fₙ^m to be the collection of n-subsets of m equipped with the symmetric difference metric. We prove that if is the minimal integer with the qth dimensional reduced homology Hq (VR (F^ₙ; 2 (n-1) ) ) being non-trivial, then rank (Hq (VR (Fₙ^m; 2 (n-1) ) ) ₈=ᵐi-2 -2 rank (Hq (VR (Fₙ^; 2 (n-1) ) ). Since the independence complex Ind (KG (n, k) ) and the Vietoris-Rips complex VR (F^2n+kₙ; 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.
Feng et al. (2024) studied this question.