More Torsion in the Homology of the Matching Complex

Abstract

A matching on a set X is a collection of pairwise disjoint subsets of X of size two. Using computers, we analyze the integral homology of the matching complex M _n , which is the simplicial complex of matchings on the set {1,…, n}. The main result is the detection of elements of order p in the homology for p ∊ {5, 7, 11, 13}. Specifically, we show that there are elements of order 5 in the homology of M _n 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 are elements of order 7 in the homology of M _n for all odd n between 23 and 41 and for n = 30. In addition, there are elements of order 11 in the homology of M₄₇ and elements of order 13 in the homology of M₆₂. Finally, we compute the ranks of the Sylow 3- and 5-subgroups of the torsion part of _d (M _n ; ) for 13 ≤ n ≤ 16; a complete description of the homology already exists for n ≤ 12. To prove the results, we use a representation-theoretic approach, examining subcomplexes of the chain complex of M _n obtained by letting certain groups act on the chain complex.

Keywords:

• Matching complex
• simplicial homology
• torsion subgroup