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

TL;DR

This paper demonstrates the presence of 5-torsion in the bottom nonvanishing homology group of the matching complex on 14 vertices, revealing an exceptional case in the homology torsion structure.

Contribution

It extends known results by showing 5-torsion in the matching complex on 14 vertices, highlighting an exceptional case in the torsion behavior of these complexes.

Findings

5-torsion exists in the homology of the matching complex on 14 vertices

For all other n, torsion is either zero or of exponent three

n=14 is an exceptional case in the torsion structure

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 $M_{14}$ 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 $n \neq 14$.