{
  "id": "math/0304280",
  "version": "v3",
  "published": "2003-04-20T00:02:16.000Z",
  "updated": "2003-11-20T15:06:46.000Z",
  "title": "Families Index for Pseudodifferential Operators on Manifolds with Boundary",
  "authors": [
    "Richard Melrose",
    "Frederic Rochon"
  ],
  "comment": "21 pages; corrected typos, changed the abstract, added a paragraph in the introduction",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "An analytic index is defined for a family of cusp pseudodifferential operators, $P_b,$ on a fibration with fibres which are compact manifolds with boundaries, provided the family is elliptic and has invertible indicial family at the boundary. In fact there is always a perturbation $Q_b$ by a family of cusp operators of order $-\\infty$ such that each $P_b+Q_b$ is invertible. Thus any elliptic family of symbols has a realization as an invertible family of cusp pseudodifferential operators, which is a form of the cobordism invariance of the index. A crucial role is played by the weak contractibility of the group of cusp smoothing operators on a compact manifold with non-trivial boundary and the associated exact sequence of classifying spaces of odd and even K-theory.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2003-11-20T15:06:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J20",
      "58J40"
    ],
    "keywords": [
      "families index",
      "cusp pseudodifferential operators",
      "compact manifold",
      "cusp operators",
      "analytic index"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......4280M"
    }
  }
}