{
  "id": "1807.11930",
  "version": "v1",
  "published": "2018-07-31T17:29:35.000Z",
  "updated": "2018-07-31T17:29:35.000Z",
  "title": "Seiberg-Witten theory on 4-manifolds with periodic ends",
  "authors": [
    "D. Veloso"
  ],
  "comment": "PhD thesis defended on December 19, 2014. http://www.theses.fr/2014AIXM4781",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this thesis we prove analytic results about a cohomotopical Seiberg-Witten theory for a Riemannian, Spin$^c$(4), 4-manifold with periodic ends, $(X, g, {\\tau})$ . Our results show that, under certain technical assumptions on $(X, g, {\\tau})$, this new version is coherent and leads to Seiberg-Witten type invariants for this new class of 4-manifolds. In the first part, using Taubes criteria for end-periodic operators, we show that for a Riemannian 4-manifold with periodic ends, $(X, g)$, verifying certain topological conditions, the Laplacian, $\\Delta_+ : L^2_2({\\Lambda^2}_+) \\rightarrow L^2({\\Lambda^2}_+)$, is a Fredholm operator. This allows us to prove a Hodge type decomposition for positively weighted Sobolev 1-forms on $(X,g)$. We also prove, assuming non-negative scalar curvature on each end and certain technical topological conditions, that the associated Dirac operator associated with an end-periodic connection (which is ASD at infinity) is Fredholm. In the second part we establish an isomorphism between the de Rham cohomology group, $H^{1}_{\\mathrm{dR}}(X,i\\mathbb{R})$ (which is a topological invariant of X) and the harmonic group intervening in the above Hodge type decomposition of the space of positively weighted 1-forms on $(X,g)$. We also prove two short exact sequences relating the gauge group of the Seiberg-Witten moduli problem and the cohomology group $H^{1}(X, 2{\\pi}i\\mathbb{Z})$. In the third part, we prove the main results: the coercivity of the Seiberg-Witten map and the compactness of the moduli space for a 4-manifold with periodic ends, $(X,g,{\\tau})$, verifying the above conditions. Finally, using the coercitivity property, we show that a Seiberg-Witten type cohomotopy invariant associated to $(X, g, {\\tau})$ can be defined",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-31T17:29:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A53",
      "53C07",
      "53C27",
      "57R57",
      "58D27",
      "47A53",
      "53C07",
      "53C27",
      "57R57",
      "58D27"
    ],
    "keywords": [
      "periodic ends",
      "seiberg-witten theory",
      "hodge type decomposition",
      "cohomology group",
      "seiberg-witten type cohomotopy invariant"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}