{
  "id": "1201.0283",
  "version": "v4",
  "published": "2011-12-31T16:27:51.000Z",
  "updated": "2013-03-07T19:36:55.000Z",
  "title": "A comparison of motivic and classical homotopy theories",
  "authors": [
    "Marc Levine"
  ],
  "comment": "4th version adds some remarks on combinatorial model categories,, makes some corrections to the appendix on symmetric products, corrects typos updates references",
  "doi": "10.1112/jtopol/jtt031",
  "categories": [
    "math.AG",
    "math.AT"
  ],
  "abstract": "Let k be an algebraically closed field of characteristic zero. Let SH(k) denote the motivic stable homotopy category of T-spectra over k and SH the classical stable homotopy category. Let c:SH -> SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In particular, c induces an isomorphism {\\pi}_n(E)-> {\\pi}_{n,0}c(E) for all spectra E. Fix an embedding of k into the complex numbers and let Re:SH(k) -> SH be the associated Betti realization. We show that the slice tower for the motivic sphere spectrum has Betti realization which is strongly convergent. This gives a spectral sequence \"of motivic origin\" converging to the homotopy groups of the classical sphere spectrum; this spectral sequence at E_2 agrees with the E_2 terms in the Adams-Novikov spectral sequence.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-03-07T19:36:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C25",
      "19E15",
      "19E08",
      "14F42",
      "55P42"
    ],
    "keywords": [
      "classical homotopy theories",
      "comparison",
      "betti realization",
      "adams-novikov spectral sequence",
      "motivic stable homotopy category"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.0283L"
    }
  }
}