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Five-torsion in the homology of the matching complex on 14 vertices. (English) Zbl 1203.05161

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 \(\mathsf {M}_{14}\) 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 \(\mathsf {M}_{n}\) when \(n\geq 13\) and \(n\neq 14\).

MSC:

05E18 Group actions on combinatorial structures
05E45 Combinatorial aspects of simplicial complexes

Software:

Homology
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References:

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