# Homology of Clique Complexes and an Application to Algebraic Geometry

# Homology of Clique Complexes and an Application to Algebraic Geometry

To any graph $G$ one can associate a simplicial complex $C(G)$, called the clique complex of $G$, whose simplices are in one-to-one correspondence with the complete subgraphs of $G$. It is thus natural to study the homology groups of $C(G)$ in connection to properties of the graph $G$. I will be interested in a special class of graphs, possessing a large group of symmetries, and particularly in Kneser graphs. The relevance of these graphs to algebraic geometry (no knowledge of which will be assumed in the talk) is that the homology of their clique complexes controls the syzygies of projective embeddings of products of projective spaces. As one of the toy examples, I will explain how computing a single homology group allows one to deduce that matrices of rank 1 are defined by the vanishing of their 2x2 minors.