On the homology of independence complexes

The independence complex Ind(G) of a graph G is the simplicial complex formed by its independent sets of vertices. We introduce a deformation of the simplicial chain complex of Ind(G) that gives rise to a spectral sequence which contains on its first page the homology groups of the independence complexes of G and various subgraphs of G, obtained by removing independent sets together with their neighborhoods. We show how this can be used to study the homology of Ind(G). Furthermore, a careful investigation of the sequence’s first page exhibits a relation between the cardinality of maximal independent sets in G and the vanishing of certain homology groups of independence complexes of subgraphs of G. We show that it holds for all paths and cycles