{
  "id": "math/0210054",
  "version": "v4",
  "published": "2002-10-04T03:52:15.000Z",
  "updated": "2003-05-04T04:32:39.000Z",
  "title": "Morse functions on the moduli space of $G_2$ structures",
  "authors": [
    "Sung Ho Wang"
  ],
  "comment": "17 pages",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Let $ \\mathfrak{M}$ be the moduli space of torsion free $ G_2$ structures on a compact 7-manifold $ M$, and let $ \\mathfrak{M}_1 \\subset \\mathfrak{M}$ be the $ G_2$ structures with volume($M$) $=1$. The cohomology map $ \\pi^3: \\mathfrak{M} \\to H^3(M, R)$ is known to be a local diffeomorphism. It is proved that every nonzero element of $ H^4(M, R) = H^3(M, R)^*$ is a Morse function on $ \\mathfrak{M}_1 $ when composed with $ \\pi^3$. When dim $H^3(M, R) = 2$, the result in particular implies $ \\pi^3$ is one to one on each connected component of $ \\mathfrak{M}$. Considering the first Pontryagin class $ p_1(M) \\in H^4(M, R)$, we formulate a compactness conjecture on the set of $ G_2$ structures of volume($M$) $=1$ with bounded $L^2$ norm of curvature, which would imply that every connected component of $ \\mathfrak{M}$ is contractible. We also observe the locus $ \\pi^3(\\mathfrak{M}_1) \\subset H^3(M, R)$ is a hyperbolic affine sphere if the volume of the torus $ H^3(M, R) / H^3(M, Z)$ is constant on $ \\mathfrak{M}_1$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2003-05-04T04:32:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C25"
    ],
    "keywords": [
      "moduli space",
      "morse function",
      "structures",
      "connected component",
      "hyperbolic affine sphere"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....10054W"
    }
  }
}