## Torsion in the Matching Complex and Chessboard Complex

**
John Shareshian and Michelle Wachs**
**Abstract:**
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.

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