Five-Torsion in the Homology of the Matching Complex on 14 Vertices

Details Five-Torsion in the Homology of the Matching Complex on 14 Vertices Five-Torsion in the Homology of the Matching Complex on 14 Vertices

2012-03-26

arxiv.org/abs/1203.5650v1

J. Algebraic Combin. 29 (2009), no. 1, 81-90

Mathematics

Combinatorics

11 pages

Scientific paper

10.1007/s10801-008-0123-6

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 $M_{14}$ 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 $M_n$ when $n \ge 13$ and $n \neq 14$.

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