Extendable periodic automorphisms of closed surfaces over the 3-sphere

TL;DR

This paper classifies all extendable periodic automorphisms of closed surfaces in the 3-sphere, including orientation-reversing cases, and shows they can be induced by automorphisms of $S^3$ on Heegaard surfaces.

Contribution

It provides a complete classification and construction of extendable automorphisms of closed surfaces in $S^3$, including orientation-reversing cases, and relates them to automorphisms of $S^3$ on Heegaard surfaces.

Findings

All extendable automorphisms of closed surfaces are classified and constructed.

Orientation-reversing extendable automorphisms are included in the classification.

Embeddings of surfaces into lens spaces are discussed.

Abstract

A periodic automorphism of a surface $\Sigma$ is said to be extendable over $S^3$ if it extends to a periodic automorphism of the pair $(S^3,\Sigma)$ for some possible embedding $\Sigma\to S^3$. We classify and construct all extendable automorphisms of closed surfaces, with orientation-reversing cases included. Moreover, they can all be induced by automorphisms of $S^3$ on Heegaard surfaces. As a by-product, the embeddings of surfaces into lens spaces are discussed.