{
  "id": "0705.0333",
  "version": "v1",
  "published": "2007-05-02T18:02:54.000Z",
  "updated": "2007-05-02T18:02:54.000Z",
  "title": "A generalization of Vassiliev's h-principle",
  "authors": [
    "Lukáš Vokřínek"
  ],
  "comment": "91 pages; PhD thesis, University of Aberdeen, 2006",
  "categories": [
    "math.AT",
    "math.DG"
  ],
  "abstract": "This thesis consists of two parts which share only a slight overlap. The first part is concerned with the study of ideals in the ring $C^\\infty(M,R)$ of smooth functions on a compact smooth manifold M or more generally submodules of a finitely generated $C^\\infty(M,R)$-module V. We define a topology on the space of all submodules of V of a fixed finite codimension d. Its main property is that it is compact Hausdorff and, in the case of ideals in the ring itself, it contains as a subspace the configuration space of d distinct unordered points in M and therefore gives a \"compactification\" of this configuration space. We present a concrete description of this space for low codimensions. The main focus is then put on the second part which is concerned with a generalization of Vassiliev's h-principle. This principle in its simplest form asserts that the jet prolongation map $j^r:C^\\infty(M,E)\\to\\Gamma(J^r(M,E))$, defined on the space of smooth maps from a compact manifold M to a Euclidean space E and with target the space of smooth sections of the jet bundle $J^r(M,E)$, is a cohomology isomorphism when restricted to certain \"nonsingular\" subsets (these are defined in terms of a certain subset $R\\subseteq J^r(M,E)$). Our generalization then puts this theorem in a more general setting of topological $C^\\infty(M,R)$-modules. As a reward we get a strengthening of this result asserting that all the homotopy fibres have zero homology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-05-02T18:02:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58K60",
      "53C23"
    ],
    "keywords": [
      "vassilievs h-principle",
      "generalization",
      "configuration space",
      "jet prolongation map",
      "compact smooth manifold"
    ],
    "tags": [
      "dissertation"
    ],
    "publication": {
      "journal": "Ph.D. Thesis",
      "year": 2007,
      "month": "May"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 91,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007PhDT........68V"
    }
  }
}