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Inequivalent Cantor sets in 𝑅³ whose complements have the same fundamental group

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  • For each Cantor set C in R³, all points of which have bounded local genus, we show that there are infinitely many inequivalent Cantor sets in R³ with the complement having the same fundamental group as the complement of C. This answers a question from Open Problems in Topology and has as an application a simple construction of nonhomeomorphic open 3-manifolds with the same fundamental group. The main techniques used are analysis of local genus of points of Cantor sets, a construction for producing rigid Cantor sets with simply connected complement, and manifold decomposition theory. The results presented give an argument that for certain groups G, there are uncountably many nonhomeomorphic open 3-manifolds with fundamental group G.
  • First published in Proceedings of the American Mathematical Society in Vol. 141 no. 8, published by the American Mathematical Society. This is the publisher’s final pdf. The published article is copyrighted by the American Mathematical Society and can be found at: http://www.ams.org/publications/journals/journalsframework/proc
  • Keywords: Cantor set, Defining sequence, End, Local genus, Open 3-manifold, Fundamental group, Rigidity
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  • Garity, D., & Repovš, D. (2013). Inequivalent Cantor sets in 𝑅³ whose complements have the same fundamental group. Proceedings of the American Mathematical Society, 141(8), 2901-2911. doi:10.1090/S0002-9939-2013-11911-8
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  • 141
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  • 8
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  • The authors were supported in part by the Slovenian Research Agency grants P1-0292-0101, J1-2057-0101 and BI-US/11-12-023. The first author was also supported in part by National Science Foundation grants DMS 0852030 and DMS 1005906.
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