{
  "id": "math/0508372",
  "version": "v1",
  "published": "2005-08-19T17:22:02.000Z",
  "updated": "2005-08-19T17:22:02.000Z",
  "title": "Cohomology of Harmonic Forms on Riemannian Manifolds With Boundary",
  "authors": [
    "Sylvain Cappell",
    "Dennis DeTurck",
    "Herman Gluck",
    "Edward Y. Miller"
  ],
  "categories": [
    "math.DG",
    "math.AT"
  ],
  "abstract": "Theorem. Let M be a compact, connected, oriented smooth Riemannian n-manifold with non-empty boundary. Then the cohomology of the complex (Harm*(M),d) of harmonic forms on M is given by the direct sum H^p(Harm*(M),d) = H^p(M;R) + H^(p-1)(M;R) for p=0,1,...,n. When M is a closed manifold, a form is harmonic if and only if it is both closed and co-closed. In this case, all the maps in the complex (Harm*(M),d) are zero, and so H^p(Harm*(M),d) = Harm^p(M) = H^p(M;R) according to the classical theorem of Hodge. By contrast, when M is connected and has non-empty boundary, it is possible for a p-form to be harmonic without being both closed and co-closed. Some of these, which are exact, although not exterior derivatives of harmonic p-1-forms, represent the \"echo\" of the ordinary p-1-dimensional cohomology within the p-dimensional harmonic cohomology that appears in the above theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-08-19T17:22:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58A12",
      "58A14",
      "14F40",
      "53C20",
      "53C21"
    ],
    "keywords": [
      "harmonic forms",
      "riemannian manifolds",
      "non-empty boundary",
      "p-dimensional harmonic cohomology",
      "oriented smooth riemannian n-manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......8372C"
    }
  }
}