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
2) We show that for almost all n, the first nontrivial
homology group of the matching complex on n vertices has
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
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.