{
  "id": "1608.08654",
  "version": "v1",
  "published": "2016-08-30T20:48:45.000Z",
  "updated": "2016-08-30T20:48:45.000Z",
  "title": "4-dimensional analogues of Dehn's lemma",
  "authors": [
    "Arunima Ray",
    "Daniel Ruberman"
  ],
  "comment": "22 pages, 22 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We investigate certain $4$-dimensional analogues of the classical $3$-dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. In particular, we show that an essential $2$-sphere $S$ in the boundary of a simply connected $4$-manifold $W$ such that $S$ is null-homotopic in $W$ need not extend to an embedding of a ball in $W$. However, if $W$ is simply connected (or more generally a $4$-manifold with abelian fundamental group) with boundary a homology sphere, then $S$ bounds a topologically embedded ball in $W$. Moreover, we give examples where such an $S$ does not bound any smoothly embedded ball in $W$. In a similar vein, we construct incompressible tori $T\\subseteq \\partial W$ where $W$ is a contractible $4$-manifold such that $T$ extends to a map of a solid torus in $W$, but not to any embedding of a solid torus in $W$. Moreover, we construct an incompressible torus $T$ in the boundary of a contractible $4$-manifold $W$ such that $T$ extends to a topological embedding of a solid torus in $W$ but no smooth embedding. As an application of our results about tori, we address a question posed by Gompf about extending certain families of diffeomorphisms of $3$-manifolds which he has recently used to construct infinite corks.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-30T20:48:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M35",
      "57N10",
      "57N13"
    ],
    "keywords": [
      "solid torus",
      "incompressible torus",
      "abelian fundamental group",
      "embedded ball",
      "dimensional dehns lemma"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}