Extending periodic automorphisms of surfaces to 3-manifolds

Let $G$ be a finite group acting on a connected compact surface $Σ$, and $M$ be an integer homology 3-sphere. We show that if each element of $G$ is extendable over $M$ with respect to a fixed embedding $Σ\rightarrow M$, then $G$ is extendable over some $M'$ which is 1-dominated by $M$. From this result, in the orientable category we classify all periodic automorphisms of closed surfaces that are extendable over the 3-sphere. The corresponding embedded surface of such an automorphism can always be a Heegaard surface.

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