Five-torsion in the homology of the matching complex on 14 vertices

Jonsson, Jakob

Journal of Algebraic Combinatorics, Tome 29 (2009) no. 1, pp. 81-90.

Résumé

Summary: J.L. Andersen proved that there is 5-torsion in the bottom nonvanishing homology group of the simplicial complex of graphs of degree at most two on seven vertices. We use this result to demonstrate that there is 5-torsion also in the bottom nonvanishing homology group of the matching complex $M _{14} \mathsf $M_14 on 14 vertices. Combining our observation with results due to Bouc and to Shareshian and Wachs, we conclude that the case $n=14$ is exceptional; for all other $n$, the torsion subgroup of the bottom nonvanishing homology group has exponent three or is zero. The possibility remains that there is other torsion than 3-torsion in higher-degree homology groups of $M _{ n} \mathsf $M_n when $n\geq 13$ and $n\neq 14$.

Détail

Keywords: keywords matching complex, simplicial homology, torsion subgroup