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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Rigid cantor sets in $ R^3$ with simply connected complement


Authors: Dennis J. Garity, Dusan Repovs and Matjaz Zeljko
Journal: Proc. Amer. Math. Soc. 134 (2006), 2447-2456
MSC (2000): Primary 54E45, 54F65; Secondary 57M30, 57N10
Published electronically: March 20, 2006
MathSciNet review: 2213719
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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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Additional Information

Dennis J. Garity
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Email: garity@math.oregonstate.edu

Dusan Repovs
Affiliation: Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, P.O. Box 2964, Ljubljana, Slovenia
Email: dusan.repovs@uni-lj.si

Matjaz Zeljko
Affiliation: Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, P.O. Box 2964, Ljubljana, Slovenia
Email: matjaz.zeljko@fmf.uni-lj.si

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08459-0
PII: S 0002-9939(06)08459-0
Keywords: Wild Cantor set, rigid set, local genus, defining sequence
Received by editor(s): September 22, 2004
Published electronically: March 20, 2006
Additional Notes: The first author was supported in part by NSF grants DMS 0139678 and DMS 0104325. The second and third authors were supported in part by MESS research program P1-0292-0101-04. All authors were supported in part by MESS grant SLO-US 2002/01 and BI-US/04-05/35.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2006 American Mathematical Society