# Symmetries of Spatial Graphs in $3$-manifolds

## TL;DR

This paper explores how graph automorphisms relate to homeomorphisms of their embeddings in 3-manifolds, revealing conditions under which symmetries are realizable and providing counterexamples in specific manifolds.

## Contribution

It establishes that all automorphisms can be realized in certain 3-manifolds, identifies limitations in orientable irreducible manifolds, and extends symmetry properties to homology spheres.

## Key findings

- Every automorphism is induced by some embedding in a connected sum of $S^2 imes S^1$
- Some automorphisms are not realizable in any embedding in orientable, closed, irreducible 3-manifolds
- Many symmetry properties extend from $S^3$ to homology spheres

## 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 Poincar\'e homology sphere, but is not induced by a homeomorphism of any embedding of $G$ in $S^3$.