{
  "id": "1203.5658",
  "version": "v2",
  "published": "2012-03-26T13:26:58.000Z",
  "updated": "2012-03-28T11:51:10.000Z",
  "title": "More Torsion in the Homology of the Matching Complex",
  "authors": [
    "Jakob Jonsson"
  ],
  "comment": "35 pages, 10 figures",
  "journal": "Experimental Mathematics 19 (2010), no. 3, 363-383",
  "doi": "10.1080/10586458.2010.10390629",
  "categories": [
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-03-28T11:51:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55U10",
      "05E25"
    ],
    "keywords": [
      "matching complex",
      "chain complex",
      "simplicial complex",
      "main result",
      "groups act"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.5658J"
    }
  }
}