{
  "id": "1201.2193",
  "version": "v2",
  "published": "2012-01-10T21:36:20.000Z",
  "updated": "2014-12-06T04:39:49.000Z",
  "title": "Arrangements of Spheres and Projective Spaces",
  "authors": [
    "Priyavrat Deshpande"
  ],
  "comment": "exposition improved, some arguments simplified, minor typos fixed, contact updated",
  "categories": [
    "math.AT",
    "math.CO"
  ],
  "abstract": "We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of intersections. We also associate a topological space: the complement of the union of tangent bundles of these subspheres in the tangent bundle of the ambient sphere. We call this space the tangent bundle complement. As in the case of hyperplane arrangements the aim of this new notion is to understand the interaction between the combinatorics of the intersections and the topology of the tangent bundle complement. In the present paper we find a closed form formula for the homotopy type of the complement and express some of its topological invariants in terms of the associated combinatorial information.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-10T21:36:20.000Z",
      "abstract": "We develop the theory of arrangements of spheres. We consider a finite collection codimension 1 spheres in a given finite dimensional sphere. To such a collection we associate two posets: the face poset and the intersection poset. We also associate a topological space to this collection. The complement of union of tangent bundles of these sub-spheres inside the tangent bundle of the ambient sphere which we call the tangent bundle complement. We find a closed form formula for the homotopy type of this complement and express some of its topological invariants in terms of the associated combinatorial information.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-06T04:39:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C35",
      "57N80",
      "05E45"
    ],
    "keywords": [
      "projective spaces",
      "arrangements",
      "tangent bundle complement",
      "finite dimensional sphere",
      "finite collection codimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.2193D"
    }
  }
}