Homology of Matching and Chessboard Complexes -Extended Abstract

Abstract:We study the topology of matching and chessboard complexes. Our main results are as follows.

1. We prove conjectures of A. Bjorner, L. Lovasz, S. T. Vrecica, and R. T. Zivaljevic 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 and n are 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 Bjorner, Lovasz, Vrecica, and Zivaljevic and is analogous to a result of G. Ziegler on chessboard complexes.

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