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**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.

### MSC:

05E18 | Group actions on combinatorial structures |

55U10 | Simplicial sets and complexes in algebraic topology |

### References:

[1] | Andersen J. L., PhD thesis, in: –Determinantal Rings Associated with Symmetric Matrices: A Counterexample.” (1992) |

[2] | Björner A., J. London Math. Soc. (2) 49 pp 25– (1994) |

[3] | DOI: 10.1016/S0021-8693(05)80055-9 · Zbl 0781.16003 |

[4] | Dong X., Electronic J. Combin. 9 pp R17– (2002) |

[5] | Hatcher A., Algebraic Topology (2002) |

[6] | DOI: 10.1016/j.jcta.2008.03.001 · Zbl 1206.55010 |

[7] | DOI: 10.1007/s10801-008-0123-6 · Zbl 1203.05161 |

[8] | DOI: 10.1017/S0305004100064768 · Zbl 0714.20008 |

[9] | Karaguezian D. B., PhD thesis, in: ”Homology of Complexes of Degree One Graphs.” (1994) |

[10] | Pilarczyk P., ”Computational Homology Program (CHomP),” (2004) |

[11] | DOI: 10.1023/A:1008728115910 · Zbl 1012.13005 |

[12] | DOI: 10.1016/j.aim.2006.10.014 · Zbl 1117.05110 |

[13] | DOI: 10.1006/jabr.1996.0317 · Zbl 0868.17015 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.