Torsion in the Matching Complex and Chessboard Complex
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. Björner, Lovász, Vrécica and {Ž}ivaljević 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 3group, improving a result of Bouc, and that the bottom nonvanishing homology of the chessboard complex $M_{n,n}$ is a 3group 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2004
 arXiv:
 arXiv:math/0409054
 Bibcode:
 2004math......9054S
 Keywords:

 Combinatorics;
 05E25;
 05E10;
 55U10
 EPrint:
 48 pages, 5 figures