## Homology of Matching and Chessboard Complexes -Extended Abstract

**
John Shareshian and Michelle L. Wachs**
**Abstract:**We study the topology of matching and chessboard
complexes. Our
main results are as follows.

- We prove conjectures of A. Bjorner, L. Lovasz, S. T. Vrecica,
and R. T. Zivaljevic on the connectivity of these complexes.
- We show that for almost all n, the first nontrivial homology group of
the matching complex on n vertices has exponent three.
- 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.
- We give a basis for the top homology group of the m-by-n chessboard
complex.
- 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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