{
  "id": "1907.03130",
  "version": "v1",
  "published": "2019-07-06T14:29:10.000Z",
  "updated": "2019-07-06T14:29:10.000Z",
  "title": "Symmetries of Spatial Graphs in Homology Spheres",
  "authors": [
    "Erica Flapan",
    "Song Yu"
  ],
  "comment": "15 pages, 10 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "This paper explores the relationship between symmetries of spatial graphs in $S^3$ and symmetries of spatial graphs in homology $3$-spheres and other $3$-manifolds. We prove that for any $3$-connected graph $G$, an automorphism $\\sigma$ is induced by a homeomorphism of some embedding of $G$ in a homology sphere if and only if $\\sigma$ is induced by a finite order homeomorphism of some embedding of $G$ in a (possibly different) homology sphere. This generalizes an analogous result for $3$-connected graphs embedded in $S^3$. On the other hand, we give an example of an automorphism of a $3$-connected graph $G$ that is realizable by a homeomorphism of some embedding of $G$ in the Poincar\\'e homology sphere, but is not realizable by a homeomorphism of any embedding of $G$ in $S^3$. Furthermore, we give an example of an automorphism of a graph $G$ which is not realizable by a homeomorphism of any embedding of $G$ in any orientable, closed, connected, irreducible $3$-manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-06T14:29:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M15",
      "05C10"
    ],
    "keywords": [
      "spatial graphs",
      "symmetries",
      "connected graph",
      "finite order homeomorphism",
      "poincare homology sphere"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}