  Topological properties of the matching complex were first studied by Bouc in
connection with Quillen complexes, and topological properties of the chessboard
complex were first studied by Garst in connection with Tits coset complexes.
Bj\"orner, Lov\'asz, Vr\'ecica and {\v{Z}}ivaljevi\'c established bounds on the
connectivity of these complexes and conjectured that these bounds are sharp. In
this paper we show that the conjecture is true by establishing the nonvanishing
of integral homology in the degrees given by these bounds. Moreover, we show
that for sufficiently large $n$, the bottom nonvanishing homology of the
matching complex $M_n$ is an elementary 3-group, improving a result of Bouc,
and that the bottom nonvanishing homology of the chessboard complex $M_{n,n}$
is a 3-group of exponent at most 9. When $n \equiv 2 \bmod 3$, the bottom
nonvanishing homology of $M_{n,n}$ is shown to be $\Z_3$. Our proofs rely on
computer calculations, long exact sequences, representation theory, and tableau
combinatorics.
