{
  "id": "2210.12483",
  "version": "v1",
  "published": "2022-10-22T15:50:40.000Z",
  "updated": "2022-10-22T15:50:40.000Z",
  "title": "The Euler characteristic, $q$-matroids, and a Möbius function",
  "authors": [
    "Trygve Johnsen",
    "Rakhi Pratihar",
    "Tovohery Hajatiana Randrianarisoa"
  ],
  "comment": "28 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We first give two new proofs of an old result that the reduced Euler characteristic of a matroid complex is equal to the M\\\"obius number of the lattice of cycles of the matroid up to the sign. The purpose has been to find a model to establish an analogous result for the case of $q$-matroids and we find a relation between the Euler characteristic of the simplicial chain complex associated to a $q$-matroid complex and the lattice of $q$-cycles of the $q$-matroid. We use this formula to find the complete homology over $\\mathbb{Z}$ of this shellable simplicial complex. We give a characterization of nonzero Euler characteristic for such order complexes. Finally, based on these results we remark why singular homology of a $q$-matroid equipped with order topology may not be effective to describe the $q$-cycles unlike the classical case of matroids.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-22T15:50:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "94B05",
      "05B35",
      "13F55"
    ],
    "keywords": [
      "möbius function",
      "matroid complex",
      "simplicial chain complex",
      "nonzero euler characteristic",
      "shellable simplicial complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}