## 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.## References

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## Additional Information

**Dennis J. Garity**- Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
- MR Author ID: 195931
- Email: garity@math.oregonstate.edu
**Dušan Repovš**- Affiliation: Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, P.O. Box 2964, Ljubljana, Slovenia
- MR Author ID: 147135
- ORCID: 0000-0002-6643-1271
- Email: dusan.repovs@uni-lj.si
**Matjaž Željko**- 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
- 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
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**134**(2006), 2447-2456 - MSC (2000): Primary 54E45, 54F65; Secondary 57M30, 57N10
- DOI: https://doi.org/10.1090/S0002-9939-06-08459-0
- MathSciNet review: 2213719