{
  "id": "1405.7087",
  "version": "v2",
  "published": "2014-05-27T23:31:22.000Z",
  "updated": "2014-12-24T04:48:33.000Z",
  "title": "$E_n$-cell attachments and a local-to-global principle for homological stability",
  "authors": [
    "Alexander Kupers",
    "Jeremy Miller"
  ],
  "comment": "36 pages, no figures. Major revision (more general result, three sections of applications)",
  "categories": [
    "math.AT"
  ],
  "abstract": "We define bounded generation for $E_n$-algebras in chain complexes and prove that for $n \\geq 2$ this property is equivalent to homological stability. Using this we prove a local-to-global principle for homological stability, which says that if an $E_n$-algebra $A$ has homological stability (or equivalently the topological chiral homology $\\int_{R^n} A$ has homology stability), then so has the topological chiral homology $\\int_M A$ of any connected non-compact manifold $M$. Using scanning, we reformulate the local-to-global homological stability principle in a way that also applies to compact manifolds. We also give several applications of our results",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-27T23:31:22.000Z",
      "abstract": "We define degreewise bounded generation for framed $E_n$-algebras in chain complexes and prove that for $n \\geq 2$ this property is equivalent to homological stability. Using this we prove a local-to-global principle for homological stability, in the sense that if a framed $E_n$-algebra $A$ has homological stability (or equivalently the topological chiral homology $\\int_{\\mathbb{R}^n} A$ has homology stability), then so has the topological chiral homology $\\int_M A$ of any open oriented connected manifold $M$.",
      "comment": "23 pages, no figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-24T04:48:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P48",
      "55R80",
      "55R40",
      "57N65"
    ],
    "keywords": [
      "homological stability",
      "local-to-global principle",
      "cell attachments",
      "topological chiral homology",
      "define degreewise bounded generation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.7087K"
    }
  }
}