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 ${\sf M}_n$, which is the simplicial complex of matchings on the set $\{1,\dots, n\}$. The main result is the detection of elements of order $p$ in the homology for $p \in \{5, 7, 11, 13\}$. Specifically, we show that there are elements of order $5$ in the homology of ${\sf M}_n$ for $n \ge 18$ and for $n \in \{14,16\}$. 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 ${\sf M}_n$ for all odd $n$ between $23$ and $41$ and for $n = 30$. In addition, there are elements of order $11$ in the homology of ${\sf M}_{47}$ and elements of order $13$ in the homology of ${\sf M}_{62}$. Finally, we compute the ranks of the Sylow 3- and 5-subgroups of the torsion part of $\tilde{H}_d({\sf M}_n; \mathbb{Z})$ for $13 \le n \le 16$; a complete description of the homology already exists for $n \le 12$. To prove the 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.