Embedding periodic maps on surfaces into those on $S^3$

TL;DR

This paper classifies when periodic maps on genus 2 surfaces can be extended to the 3-sphere, revealing that all even-genus maximum order maps are extendable, contrasting with orientation-preserving cases.

Contribution

It explicitly determines extendability of all periodic maps on genus 2 surfaces and provides a detailed decomposition into symmetries, offering new insights into surface embeddings in $S^3$.

Findings

All periodic maps on $oldsymbol{ ext{genus 2}}$ surfaces are classified.

Maximum order maps on even genus surfaces are extendable.

Explicit symmetry decompositions of periodic maps are provided.

Abstract

Call a periodic map $h$ on the closed orientable surface $\Sigma_g$ extendable if $h$ extends to a periodic map over the pair $(S^3, \Sigma_g)$ for possible embeddings $e: \Sigma_g\to S^3$. We determine the extendabilities for all periodical maps on $\Sigma_2$. The results involve various orientation preserving/reversing behalves of the periodical maps on the pair $(S^3, \Sigma_g)$. To do this we first list all periodic maps on $\Sigma_2$, and indeed we exhibit each of them as a composition of primary and explicit symmetries, like rotations, reflections and antipodal maps, which itself should be an interesting piece. A by-product is that for each even $g$, the maximum order periodic map on $\Sigma_g$ is extendable, which contrasts sharply to the situation in orientation preserving category.