{
  "id": "1702.04417",
  "version": "v1",
  "published": "2017-02-14T23:10:58.000Z",
  "updated": "2017-02-14T23:10:58.000Z",
  "title": "A splitting theorem for the Seiberg-Witten invariant of a homology $S^1 \\times S^3$",
  "authors": [
    "Jianfeng Lin",
    "Daniel Ruberman",
    "Nikolai Saveliev"
  ],
  "comment": "75 pages, 1 figure",
  "categories": [
    "math.GT",
    "math.AP",
    "math.DG"
  ],
  "abstract": "We study the Seiberg-Witten invariant $\\lambda_{\\rm{SW}} (X)$ of smooth spin $4$-manifolds $X$ with integral homology of $S^1\\times S^3$ defined by Mrowka, Ruberman, and Saveliev as a signed count of irreducible monopoles amended by an index-theoretic correction term. We prove a splitting formula for this invariant in terms of the Fr{\\o}yshov invariant $h(X)$ and a certain Lefschetz number in the reduced monopole Floer homology of Kronheimer and Mrowka. We apply this formula to obstruct existence of metrics of positive scalar curvature on certain 4-manifolds, and to exhibit new classes of integral homology $3$-spheres of Rohlin invariant one which have infinite order in the homology cobordism group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-14T23:10:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R57",
      "57R58",
      "57M27",
      "53C21",
      "58J28",
      "58J35"
    ],
    "keywords": [
      "seiberg-witten invariant",
      "splitting theorem",
      "integral homology",
      "index-theoretic correction term",
      "reduced monopole floer homology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 75,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}