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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Distinguishing Bing-Whitehead Cantor sets


Authors: Dennis Garity, Dušan Repovš, David Wright and Matjaž Željko
Journal: Trans. Amer. Math. Soc. 363 (2011), 1007-1022
MSC (2000): Primary 54E45, 54F65; Secondary 57M30, 57N10
Published electronically: September 17, 2010
MathSciNet review: 2728594
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Abstract: Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were nonstandard (wild), but still had a simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions greater than three. These Cantor sets in $ S^{3}$ are constructed by using Bing or Whitehead links as stages in defining sequences. Ancel and Starbird, and separately Wright, characterized the number of Bing links needed in such constructions so as to produce Cantor sets. However it was unknown whether varying the number of Bing and Whitehead links in the construction would produce nonequivalent Cantor sets. Using a generalization of the geometric index, and a careful analysis of three dimensional intersection patterns, we prove that Bing-Whitehead Cantor sets are equivalently embedded in $ S^3$ if and only if their defining sequences differ by some finite number of Whitehead constructions. As a consequence, there are uncountably many nonequivalent such Cantor sets in $ S^{3}$ constructed with genus one tori and with a simply connected complement.


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Additional Information

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

Dušan Repovš
Affiliation: Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana, P.O. Box 2964, Ljubljana, Slovenia 1001
Email: dusan.repovs@guest.arnes.si

David Wright
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: wright@math.byu.edu

Matjaž Željko
Affiliation: Institute of Mathematics, Physics and Mechanics, Faculty of Mathematics and Physics, University of Ljubljana, P.O.Box 2964, Ljubljana, Slovenia
Email: matjaz.zeljko@fmf.uni-lj.si

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05175-X
PII: S 0002-9947(2010)05175-X
Keywords: Cantor set, wild Cantor set, Bing link, Whitehead link, defining sequence
Received by editor(s): October 19, 2008
Received by editor(s) in revised form: July 1, 2009
Published electronically: September 17, 2010
Article copyright: © Copyright 2010 American Mathematical Society