Homology of Matching and Chessboard Complexes

John Shareshian and Michelle Wachs

To appear at Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe, Arizona (USA), May 20-26, 2001


We study the topology of matching and chessboard complexes. Our main results are as follows. 1) We prove conjectures of A. Bj\"orner, L. Lov\'asz, S. Vre\'cica and R. \v Zivaljevi\'c 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 many pairs m,n. 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 Bj\"orner, Lov\'asz, Vre\'cica and \v Zivaljevi\'c and is analogous to a result of G. Ziegler on chessboard complexes.

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