Symmetries of Spatial Graphs in Homology Spheres

Erica Flapan, Song Yu

Published 2019-07-06Version 1

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.