Abstract
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≥13 and n≠14.
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Research of J. Jonsson was supported by European Graduate Program “Combinatorics, Geometry, and Computation”, DFG-GRK 588/2.
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Jonsson, J. Five-torsion in the homology of the matching complex on 14 vertices. J Algebr Comb 29, 81–90 (2009). https://doi.org/10.1007/s10801-008-0123-6
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DOI: https://doi.org/10.1007/s10801-008-0123-6