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 F₍^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 Hₐ (VR (F₍^l;2 (n-1) ) ) being non-trivial, then rank (Hₐ (VR (F₍^m;2 (n-1) ) ) ₈=₋^mi-2l-2 rank (Hₐ (VR (F₍^l;2 (n-1) ) ). Since we know the independence complex Ind (KG (n, k) ) and the Vietoris-Rips complex VR (F₍^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.