Homology of Bounded Degree Graph Complexes

To every finite collection of graphs that is closed under removal of edges, one can associate an abstract simplicial complex whose faces are the edge sets of the graphs in the collection. Graph complexes have provided an important link between combinatorics and algebra, topology and geometry. In this talk we will discuss the simplicial complex associated with the collection of subgraphs of a graph G whose maximum vertex degree is at most b. Some important special cases which have arisen in various contexts in the recent literature are the matching complex (G is a complete graph and b=1) and the chessboard complex (G is a complete bipartite graph and b=1). Topological properties of the matching complex were first examined by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first examined by Garst in connection with Coxeter complexes. In this talk I will survey old and new results on the homology of bounded degree graph complexes. I will discuss the roles played by some classical symmetric function identities of Littlewood, the representation theory of symmetric and general linear groups, and the Robinson-Schensted correspondence.