Torsion in the matching complex and chessboard complex

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. Björner, Lovász, Vrećica and Živaljević 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 ≡ 2 mod 3, the bottom nonvanishing homology of M_{n, n} is shown to be Z₃. Our proofs rely on computer calculations, long exact sequences, representation theory, and tableau combinatorics.