{
  "id": "1103.5457",
  "version": "v2",
  "published": "2011-03-28T19:50:25.000Z",
  "updated": "2011-05-17T13:19:55.000Z",
  "title": "Combinatorics of embeddings",
  "authors": [
    "Sergey A. Melikhov"
  ],
  "comment": "49 pages, 1 figure. Minor improvements in v2 (subsection 4.C on transforms of dichotomial spheres reworked to include more details; subsection 2.D \"Algorithmic issues\" added, etc)",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere; there exist precisely two 3-dimensional dichotomial cell complexes, and their 1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their 1-skeleta all but one graphs of the Petersen family. In higher dimensions n>2, we observe that in order to characterize those compact n-polyhedra that embed in S^{2n} in terms of finitely many \"prohibited minors\", it suffices to establish finiteness of the list of all (n-1)-connected n-dimensional finite cell complexes that do not embed in S^{2n} yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the n-skeleta of (2n+1)-dimensional dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell complexes include (apart from the three joins of the i-skeleta of (2i+2)-simplices) at least ten non-simplicial complexes.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-05-17T13:19:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q35",
      "57M15",
      "57M20",
      "05C83",
      "05C10"
    ],
    "keywords": [
      "dichotomial cell complexes",
      "cell-like combinatorial quotients",
      "kuratowski graph planarity criterion",
      "embeddings",
      "n-dimensional finite cell complexes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.5457M"
    }
  }
}