More Torsion in the Homology of the Matching Complex

Using computers, we analyze the integral homology of the matching complex Mn, which is the simplicial complex of matchings in the complete graph on n vertices. Our main result is the detection of p-torsion in the homology for p ∈ {5, 7, 11, 13}. Specifically, we show that there is nonvanishing 5-torsion in the homology of Mn for n ≥ 18 and for n ∈ {14, 16}. The only previously known value was n = 14, and in this particular case we have a new computer-free proof. Moreover, we show that there is 7-torsion in the homology of Mn for all odd n between 23 and 41 and for n = 30. In addition, there is 11-torsion in the homology of M47 and 13-torsion in the homology of M62. Finally, we compute the ranks of the 3and 5-subgroups of H̃d(Mn; Z) for 13 ≤ n ≤ 16; a complete description of the homology already exists for n ≤ 12. To prove our results, we use a representation-theoretic approach, examining subcomplexes of the chain complex of Mn obtained by letting certain groups act on the chain complex.