{
  "id": "2206.05721",
  "version": "v1",
  "published": "2022-06-12T12:03:16.000Z",
  "updated": "2022-06-12T12:03:16.000Z",
  "title": "On the homology of the noncrossing partition lattice and the Milnor fibre",
  "authors": [
    "Yang Zhang"
  ],
  "comment": "31 pages",
  "categories": [
    "math.CO",
    "math.AG",
    "math.AT",
    "math.RA",
    "math.RT"
  ],
  "abstract": "Let $\\mathcal{L}$ be the noncrossing partition lattice associated to a finite Coxeter group $W$. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of $\\mathcal{L}$. We define a multiplicative structure on the Whitney homology of $\\mathcal{L}$ in terms of the basis, and the resulting algebra has similarities to the Orlik-Solomon algebra. As an application, we obtain four chain complexes which compute the integral homology of the Milnor fibre of the reflection arrangement of $W$, the Milnor fibre of the discriminant of $W$, the hyperplane complement of $W$ and the Artin group of type $W$, respectively. We also tabulate some computational results on the integral homology of the Milnor fibres.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-12T12:03:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "52C35",
      "05E14"
    ],
    "keywords": [
      "noncrossing partition lattice",
      "milnor fibre",
      "integral homology",
      "finite coxeter group",
      "construct explicit bases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}