More Torsion in the Homology of the Matching Complex

TL;DR

This paper investigates the torsion elements in the homology of the matching complex, discovering new torsion elements of orders 5, 7, 11, and 13 for various n, using computational and representation-theoretic methods.

Contribution

It provides new computer-assisted proofs and extends known results on torsion in the homology of matching complexes, including elements of orders 5, 7, 11, and 13.

Findings

Elements of order 5 in homology for n ≥ 18 and n=14,16.

Elements of order 7 in homology for all odd n between 23 and 41, and n=30.

Elements of order 11 in homology for n=47, and order 13 for n=62.

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 $p \in \{5,7,11,13\}$. Specifically, we show that there are elements of order 5 in the homology of $M_n$ 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 $M_n$ for all odd $n$ between 23 and 41 and for $n=30$. In addition, there are elements of order 11 in the homology of $M_{47}$ and elements of order 13 in the homology of $M_{62}$. Finally, we compute the ranks of the Sylow 3- and 5-subgroups of the torsion part of $H_d(M_n;Z)$ 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 $M_n$ obtained by letting certain groups act on the chain complex.