More torsion in the homology of the matching complex. (English) Zbl 1247.05271

Summary: 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 \(\{1,\dots, n\}\). 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 \(M_n\) for \(n \geq 18\) 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 \(\tilde{H}_d(M_n; \mathbb{Z})\) for \(13 \leq n \leq 16\); a complete description of the homology already exists for \(n \leq 12\). 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.


05E18 Group actions on combinatorial structures
55U10 Simplicial sets and complexes in algebraic topology
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