Abstract
We construct uncountably many simply connected open 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed manifold has the property that any self homeomorphism of the manifold (which necessarily extends to a homeomorphism of the ends) fixes the ends pointwise. These manifolds are complements of rigid generalized Bing–Whitehead (BW) Cantor sets. Previous examples of rigid Cantor sets with simply connected complement in \(R^{3}\) had infinite genus and it was an open question as to whether finite genus examples existed. The examples here exhibit the minimum possible genus, genus one. These rigid generalized BW Cantor sets are constructed using variable numbers of Bing and Whitehead links. Our previous result with Željko determining when BW Cantor sets are equivalently embedded in \(R^{3}\) extends to the generalized construction. This characterization is used to prove rigidity and to distinguish the uncountably many examples.
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Acknowledgments
The first author was supported in part by the National Science Foundation grant DMS0852030. The first and second authors were supported in part by the Slovenian Research Agency grant BI-US/11-12-023. The first and third authors were supported in part by the National Science Foundation grant DMS1005906. The second author was supported in part by the Slovenian Research Agency grants P1-0292-0101 and J1-2057-0101. The authors would also like to thank the referees for helpful comments and suggestions.
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Garity, D., Repovš, D. & Wright, D. Simply connected open 3-manifolds with rigid genus one ends. Rev Mat Complut 27, 291–304 (2014). https://doi.org/10.1007/s13163-013-0117-3
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DOI: https://doi.org/10.1007/s13163-013-0117-3
Keywords
- Open 3-manifold
- Rigidity
- Genus one manifold end
- Wild Cantor set
- Bing link
- Whitehead link
- Defining sequence
- Bing–Whitehead Cantor set