{
  "id": "1305.6100",
  "version": "v4",
  "published": "2013-05-27T03:34:47.000Z",
  "updated": "2015-07-12T18:34:05.000Z",
  "title": "The homology of $\\mathrm{tmf}$",
  "authors": [
    "Akhil Mathew"
  ],
  "comment": "30 pages; 3 figures. Submitted version. Comments always welcome",
  "categories": [
    "math.AT",
    "math.AG"
  ],
  "abstract": "We compute the mod $2$ homology of the spectrum $\\mathrm{tmf}$ of topological modular forms by proving a 2-local equivalence $\\mathrm{tmf} \\wedge DA(1) \\simeq \\mathrm{tmf}_1(3) \\simeq BP\\left \\langle 2\\right\\rangle$, where $DA(1)$ is an eight cell complex whose cohomology \"doubles\" the subalgebra $\\mathcal{A}(1)$ of the Steenrod algebra generated by $\\mathrm{Sq}^1$ and $\\mathrm{Sq}^2$. To do so, we give, using the language of stacks, a modular description of the elliptic homology of $DA(1)$ via level three structures. We briefly discuss analogs at odd primes and recover the stack-theoretic description of the Adams-Novikov spectral sequence for $\\mathrm{tmf}$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-11-20T14:40:15.000Z",
      "abstract": "We compute the mod 2 homology of the spectrum $\\mathrm{tmf}$ of topological modular forms by proving a 2-local equivalence $\\mathrm{tmf} \\wedge DA(1) \\simeq BP\\left\\langle 2\\right\\rangle$, where $DA(1)$ is an eight cell complex whose cohomology \"doubles\" the subalgebra $\\mathcal{A}(1)$ of the Steenrod algebra generated by $\\mathrm{Sq}^{1}$ and $\\mathrm{Sq}^{2}$. To do so, we give, with use of the language of stacks, a modular description of the elliptic homology of $DA(1)$ via level three structures. We briefly discuss analogs at odd primes and recover the stack-theoretic description of the Adams-Novikov spectral sequence for $\\mathrm{tmf}$.",
      "comment": "50 pages; 5 figures. Comments welcome. 5/30/13: several minor corrections and additional remarks added; 11/20/13: fixed LaTeX errors in abstract",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-07-12T18:34:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "adams-novikov spectral sequence",
      "cell complex",
      "modular description",
      "topological modular forms",
      "elliptic homology"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.6100M"
    }
  }
}