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Symmetries of spatial graphs in 3-manifolds

Volume 255 / 2021

Erica Flapan, Song Yu Fundamenta Mathematicae 255 (2021), 289-308 MSC: Primary 57M15, 05E18; Secondary 57M60, 05C10, 57S25. DOI: 10.4064/fm798-5-2021 Published online: 7 September 2021

Abstract

We consider when automorphisms of a graph can be induced by homeomorphisms of embeddings of the graph in a $3$-manifold. In particular, we prove that every automorphism of a graph is induced by a homeomorphism of some embedding of the graph in a connected sum of one or more copies of $S^2\times S^1$, yet there exist automorphisms which are not induced by a homeomorphism of any embedding of the graph in any orientable, closed, connected, irreducible $3$-manifold. We also prove that for any $3$-connected graph $G$, if an automorphism $\sigma $ is induced by a homeomorphism of an embedding of $G$ in an irreducible $3$-manifold $M$, then $G$ can be embedded in an orientable, closed, connected $3$-manifold $M’$ such that $\sigma $ is induced by a finite order homeomorphism of $M’$, though this is not true for graphs which are not $3$-connected. Finally, we show that many symmetry properties of graphs in $S^3$ hold for graphs in homology spheres, yet we give an example of an automorphism of a graph $G$ that is induced by a homeomorphism of some embedding of $G$ in the Poincaré homology sphere, but is not induced by a homeomorphism of any embedding of $G$ in $S^3$.

Authors

  • Erica FlapanDepartment of Mathematics
    Pomona College
    Claremont, CA 91711, U.S.A.
    e-mail
  • Song YuDepartment of Mathematics
    Columbia University
    New York, NY 10027, U.S.A.
    e-mail

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