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Mathematics > Geometric Topology

arXiv:1803.05116 (math)
[Submitted on 14 Mar 2018 (v1), last revised 25 Apr 2019 (this version, v3)]

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

Authors:Kenta Funayoshi, Yuya Koda
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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.
Comments: 23 pages, 16 figures; typos and errors corrected
Subjects: Geometric Topology (math.GT)
MSC classes: 57M60, 57S25
Cite as: arXiv:1803.05116 [math.GT]
  (or arXiv:1803.05116v3 [math.GT] for this version)
  https://doi.org/10.48550/arXiv.1803.05116

Submission history

From: Yuya Koda [view email]
[v1] Wed, 14 Mar 2018 02:56:23 UTC (42 KB)
[v2] Sat, 17 Mar 2018 08:20:15 UTC (42 KB)
[v3] Thu, 25 Apr 2019 23:26:00 UTC (191 KB)
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