Topology of matching, chessboard, and general bounded degree graph complexes

Abstract

Abstract. We survey results and techniques in the topological study of simplicial complexes of (di-, multi-, hyper-)graphs whose node degrees are bounded from above. These complexes have arisen is a variety of contexts in the literature. The most wellknown examples are the matching complex and the chessboard complex. The topics covered here include computation of Betti numbers, representations of the symmetric group on rational homology, torsion in integral homology, homotopy properties, and connections with other fields.

Decompositions and Connectivity of Matching and Chessboard Complexes

“…Since then they have turned out to be of importance in various mathematical contexts within algebra [11], combinatorics [14], discrete and computational geometry [17], representation theory [7] and topology [1]; see the recent survey [15] for a detailed historic account and a nice exposition of the main results and techniques in the study of matching and chessboard complexes, as well as for further references. Their hypergraph analogues were introduced and studied mainly with respect to their connectivity properties in [5].…”

“…This was proved for the chessboard complex M m,n by Ziegler [16], who showed that the respective skeleton has the even stronger property of being vertex decomposable, and for the matching complex M n by Shareshian and Wachs [12], who describe an explicit shelling order of the facets of the (n − 2)/3 -skeleton of M n . The question whether this skeleton is vertex decomposable was raised by Wachs [15,Problem 5.4].…”

Connectivity at infinity for braid groups on complete graphs

“…Björner, Lovász, Vrećica andŽivaljević proved a connectivity bound [5], which was eventually shown to be sharp (see [18] and [19]). The result is: …”

Four-Cycled Graphs with Topological Applications

“…In particular, the clique complexes of the complements of the line graphs of bipartite graphs are known as chessboard complexes in combinatorics which are quite well-studied objects in representation theory and commutative algebra [14]. We first verify that any chessboard complex can be obtained in this way.…”