Extended Abstract HOMOLOGY OF MATCHING AND CHESSBOARD COMPLEXES

We study the topology of matching and chessboard complexes. Our main results are as follows. 1. We prove conjectures of A. Björner, L. Lovász, S. T. Vrećica, and R. T. Živaljević on the connectivity of these complexes. 2. We show that for almost all n, the first nontrivial homology group of the matching complex on n vertices has exponent three. 3. We prove similar but weaker results on the exponent of the first nontrivial homology group of the m-by-n chessboard complex for all pairs m < n for which m is sufficiently large and n−m is sufficiently small. 4. We give a basis for the top homology group of the m-by-n chessboard complex. 5. We prove that a certain skeleton of the matching complex is shellable. This result answers a question of Björner, Lovász, Vrećica, and Živaljević and is analogous to a result of G. Ziegler on chessboard complexes. A matching on vertex set V is a graph in which each vertex is contained in at most one edge. If G = (V,E) is a graph then the collection of all subgraphs (V,E′) of G which are matchings determines a simplicial complex M(G), as follows. The vertices of M(G) are (in correspondence with) the edges of G, and the k-dimensional faces of M(G) are the subgraphs of G which are matchings with k + 1 edges. If G is the complete graph on vertex set [n] := {1, 2, . . . , n} for some positive integer n then we write Mn for M(G). Similarly, if G is the complete bipartite graph with parts [m] and [n]′ := {1′, 2′, . . . , n′} for positive integers m,n then we write Mm,n for M(G). The complexes Mn are called matching complexes. Topological properties of these complexes were first examined by S. Bouc in [Bo], in connection with the Quillen complexes at the prime 2 for the symmetric groups Sn. One of Bouc’s main results shows that the homology of Mn with complex coefficients behaves quite nicely. Note that the natural action of Sn determines an action of Sn on Mn, which in turn determines a representation of Sn on each reduced homology group of Mn. As is well known, the irreducible complex representations of Sn are indexed by partitions of n. For a partition λ, let Sλ be the irreducible representation corresponding to λ, let λ′ be the partition conjugate to λ and let d(λ) be the sidelength of the largest square which fits inside the Young diagram of λ (see [MacD],[Sa] or [St, Chapter 7] for definitions). Bouc’s result is as follows. Theorem 1 ([Bo, Proposition 5]). For all i ≥ 1 and all n ≥ 2, there is an isomorphism H̃i−1(Mn,C) ∼= ⊕ λ : λ ` n, λ = λ′, d(λ) = n− 2i S