{
  "id": "2002.02326",
  "version": "v1",
  "published": "2020-02-06T16:07:36.000Z",
  "updated": "2020-02-06T16:07:36.000Z",
  "title": "Corks, involutions, and Heegaard Floer homology",
  "authors": [
    "Irving Dai",
    "Matthew Hedden",
    "Abhishek Mallick"
  ],
  "comment": "67 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "Building on the algebraic framework developed by Hendricks, Manolescu, and Zemke, we introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. Unlike previous approaches, we do not utilize any closed 4-manifold topology or contact topology. Instead, we adapt the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification $\\Theta^{\\tau}_{\\mathbb{Z}}$ of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits a $\\mathbb{Z}^\\infty$-subgroup of strongly non-extendable corks. The group $\\Theta^{\\tau}_{\\mathbb{Z}}$ can also be viewed as a refinement of the bordism group of diffeomorphisms. Using our invariants, we furthermore establish several new families of corks and prove that various known examples are strongly non-extendable. Our main computational tool is a monotonicity theorem which constrains the behavior of our invariants under equivariant negative-definite cobordisms, and an explicit method of constructing such cobordisms via equivariant surgery.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-06T16:07:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57R58"
    ],
    "keywords": [
      "involution",
      "involutive heegaard floer homology",
      "homology cobordism group",
      "main computational tool",
      "equivariant negative-definite cobordisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 67,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}