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

DOI:
https://doi.org/10.1090/S0002-9947-2010-05175-X

Published electronically:
September 17, 2010

MathSciNet review:
2728594

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Abstract | References | Similar Articles | Additional Information

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 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 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 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:
https://doi.org/10.1090/S0002-9947-2010-05175-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