Extending automorphisms of the genus-2 surface over the 3-sphere

TL;DR

This paper characterizes when automorphisms of genus-2 surfaces can be extended over the 3-sphere, showing they must extend over embeddings where the surface bounds genus-2 handlebodies on both sides.

Contribution

It proves that extendable automorphisms of genus-2 surfaces extend over embeddings with specific handlebody boundary conditions, utilizing classification of essential annuli.

Findings

Automorphisms extend over embeddings with genus-2 handlebodies on both sides

Classification of essential annuli is key to the proof

Provides criteria for extendability over the 3-sphere

Abstract

An automorphism $f$ of a closed orientable surface $\Sigma$ is said to be extendable over the 3-sphere $S^3$ if $f$ extends to an automorphism of the pair $(S^3, \Sigma)$ with respect to some embedding $\Sigma \hookrightarrow S^3$. We prove that if an automorphism of a genus-2 surface $\Sigma$ is extendable over $S^3$, then $f$ extends to an automorphism of the pair $(S^3, \Sigma)$ with respect to an embedding $\Sigma \hookrightarrow S^3$ such that $\Sigma$ bounds genus-2 handlebodies on both sides. The classification of essential annuli in the exterior of genus-2 handlebodies embedded in $S^3$ due to Ozawa and the second author plays a key role.