Rigid cantor sets in 𝑅³ with simply connected complement

Garity, Dennis; Repovš, Dušan; Željko, Matjaž

doi:10.1090/S0002-9939-06-08459-0

Rigid cantor sets in $R^3$ with simply connected complement
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by Dennis J. Garity, Dušan Repovš and Matjaž Željko PDF

Proc. Amer. Math. Soc. 134 (2006), 2447-2456 Request permission

Abstract:

We prove that there exist uncountably many inequivalent rigid wild Cantor sets in $R^{3}$ with simply connected complement. Previous constructions of wild Cantor sets in ${R}^{3}$ with simply connected complement, in particular the Bing- Whitehead Cantor sets, had strong homogeneity properties. This suggested it might not be possible to construct such sets that were rigid. The examples in this paper are constructed using a generalization of a construction of Skora together with a careful analysis of the local genus of points in the Cantor sets.

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