On the homology of independence complexes
Abstract
The independence complex Ind(G) of a graph G is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of Ind(G) that gives rise to a double complex with trivial homology. Filtering this double complex in the right direction induces a spectral sequence that converges to zero and contains on its first page the homology of the independence complexes of G and various subgraphs of G, obtained by removing independent sets and their neighborhoods from G. It is shown that this spectral sequence may 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 the independence complexes of some subgraphs of G. This relation is shown to hold for all paths and cyclic graphs.