{
  "id": "1801.06410",
  "version": "v1",
  "published": "2018-01-19T14:12:26.000Z",
  "updated": "2018-01-19T14:12:26.000Z",
  "title": "The $\\mathcal L_B$-cohomology on compact torsion-free $\\mathrm{G}_2$ manifolds and an application to `almost' formality",
  "authors": [
    "Ki Fung Chan",
    "Spiro Karigiannis",
    "Chi Cheuk Tsang"
  ],
  "comment": "40 pages, 8 figures in colour",
  "categories": [
    "math.DG"
  ],
  "abstract": "We study a cohomology theory $H^{\\bullet}_{\\varphi}$, called the $\\mathcal L_B$-cohomology, on compact torsion-free $\\mathrm{G}_2$-manifolds. We show that $H^k_{\\varphi} \\cong H^k_{\\mathrm{dR}}$ for $k \\neq 3, 4$, but that $H^k_{\\varphi}$ is infinite-dimensional for $k = 3,4$. Nevertheless there is a canonical injection $H^k_{\\mathrm{dR}} \\to H^k_{\\varphi}$. The $\\mathcal L_B$-cohomology also satisfies a Poincar\\'e duality induced by the Hodge star. The establishment of these results requires a delicate analysis of the interplay between the exterior derivative $\\mathrm{d}$ and the derivation $\\mathcal L_B$, and uses both Hodge theory and the special properties of $\\mathrm{G}_2$-structures in an essential way. As an application of our results, we prove that compact torsion-free $\\mathrm{G}_2$-manifolds are `almost formal' in the sense that most of the Massey triple products necessarily must vanish.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-19T14:12:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C25",
      "53C29"
    ],
    "keywords": [
      "compact torsion-free",
      "application",
      "cohomology theory",
      "massey triple products necessarily",
      "delicate analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}