{
  "id": "0905.2591",
  "version": "v3",
  "published": "2009-05-15T18:27:52.000Z",
  "updated": "2009-12-24T16:43:42.000Z",
  "title": "On the Betti numbers of a loop space",
  "authors": [
    "Samson Saneblidze"
  ],
  "comment": "11 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "Let $A$ be a special homotopy G-algebra over a commutative unital ring $\\Bbbk$ such that both $H(A)$ and $\\operatorname{Tor}_{i}^{A}(\\Bbbk,\\Bbbk)$ are finitely generated $\\Bbbk$-modules for all $i$, and let $\\tau_{i}(A)$ be the cardinality of a minimal generating set for the $\\Bbbk$-module $\\operatorname{Tor}_{i}^{A}(\\Bbbk,\\Bbbk).$ Then the set ${\\tau_{i}(A)} $ is unbounded if and only if $\\tilde{H}(A)$ has two or more algebra generators. When $A=C^{\\ast}(X;\\Bbbk)$ is the simplicial cochain complex of a simply connected finite $CW$-complex $X,$ there is a similar statement for the \"Betti numbers\" of the loop space $\\Omega X.$ This unifies existing proofs over a field $\\Bbbk$ of zero or positive characteristic.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-12-24T16:43:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P35",
      "55U20",
      "55S30"
    ],
    "keywords": [
      "loop space",
      "betti numbers",
      "special homotopy g-algebra",
      "simplicial cochain complex",
      "algebra generators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.2591S"
    }
  }
}