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Extending automorphisms of the genus-2 surface over the 3-sphere (1803.05116v3)
Published 14 Mar 2018 in math.GT
Abstract: An automorphism $f$ of a closed orientable surface $\Sigma$ is said to be extendable over the 3-sphere $S3$ if $f$ extends to an automorphism of the pair $(S3, \Sigma)$ with respect to some embedding $\Sigma \hookrightarrow S3$. We prove that if an automorphism of a genus-2 surface $\Sigma$ is extendable over $S3$, then $f$ extends to an automorphism of the pair $(S3, \Sigma)$ with respect to an embedding $\Sigma \hookrightarrow S3$ such that $\Sigma$ bounds genus-2 handlebodies on both sides. The classification of essential annuli in the exterior of genus-2 handlebodies embedded in $S3$ due to Ozawa and the second author plays a key role.