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 $p\in \{5,7,11,13\}$. 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.

arXiv comment35 pages, 10 figures

References in corpus (2)

• Exact Sequences for the Homology of the Matching Complex
• Five-Torsion in the Homology of the Matching Complex on 14 Vertices

Cited by in corpus (3)

• Filtered matchings and simplicial complexes
• On the 3-torsion Part of the Homology of the Chessboard Complex
• The graph minor theorem in topological combinatorics