Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

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

Author(s): Dennis J. Garity; Dusan Repovs; Matjaz Zeljko
Journal: Proc. Amer. Math. Soc. 134 (2006), 2447-2456.
MSC (2000): Primary 54E45, 54F65; Secondary 57M30, 57N10
Posted: March 20, 2006
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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:

[AS]
F. D. Ancel and M. P. Starbird, The shrinkability of Bing-Whitehead decompositions, Topology 28 (1989), no. 3, 291-304. MR 1014463 (90g:57014)

[Ar]
S. Armentrout, Decompositions of $ \mathbf{E}^3$ with a compact 0-dimensional set of nondegenerate elements, Trans. Amer. Math. Soc. 123 (1966), 165-177 MR 0195074 (33:3279)

[Ba]
A. Babich, Scrawny Cantor sets are not definable by tori, Proc. Amer. Math. Soc. 115 (1992), 829-836. MR 1106178 (92i:57012)

[Bi]
R. H. Bing, Tame Cantor sets in $ R^3$, Pacific J. Math. 11 (1961), 435-446. MR 0130679 (24:A539)

[Bl]
W. A. Blankinship, Generalization of a construction of Antoine, Ann. of Math. (2) 53 (1951), 276-297. MR 0040659 (12:730c)

[Da]
R. J. Daverman, Decompositions of Manifolds, Pure and Appl. Math. 124, Academic Press, Orlando, 1986. MR 0872468 (88a:57001)

[Ma]
J. M. Martin, A rigid sphere, Fund. Math. 59 (1966), 117-121. MR 0224075 (36:7122)

[DO]
D. G. DeGryse and R. P. Osborne, A wild Cantor set in $ E\sp{n}$ with simply connected complement, Fund. Math. 86 (1974), 9-27. MR 0375323 (51:11518)

[Ki]
A. Kirkor, Wild 0-dimensional sets and the fundamental group, Fund. Math. 45 (1958), 228-236. MR 0102783 (21:1569)

[Os]
R. P. Osborne, Embedding Cantor sets in a manifold, Part I: Tame Cantor sets in $ R^n$, Michigan Math. J. 13 (1966), 57-63. MR 0187225 (32:4678)

[Sh]
R. B. Sher, Concerning wild Cantor sets in $ R^3$, Proc. Amer. Math. Soc. 19 (1968), 1195-1200. MR 0234438 (38:2755)

[Sl]
A. C. Shilepsky, A rigid Cantor set in $ \mathbb{E}^3$, Bull. Acad. Polon. Sci. 22 (1974), 223-224. MR 0345110 (49:9849)

[Sk]
R. Skora, Cantor sets in $ S^3$ with simply connected complements, Topol. Appl.  24 (1986), 181-188. MR 0872489 (87m:57009)

[Wr1]
D. G. Wright, Cantor sets in 3-manifolds, Rocky Mountain J. Math.  9 (1979), 377-383. MR 0519952 (80j:57011)

[Wr2]
D. G. Wright, Rigid sets in manifolds, Geometric and Algebraic Topology, Banach Center Publ. 18, PWN, Warsaw 1986, 161-164. MR 0925863 (89a:57019)

[Wr3]
D. G. Wright, Rigid sets in $ E^n$, Pacific J. Math. 121 (1986), 245-256. MR 0815045 (87b:57011)

[Wr4]
D. G. Wright, Bing-Whitehead Cantor sets, Fund. Math. 132 (1989), 105-116. MR 1002625 (90d:57020)

[Ze]
M. Zeljko, Genus of a Cantor Set, Rocky Mountain J. Math., 35 (2005), no. 1. MR 2117612

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54E45, 54F65, 57M30, 57N10

Retrieve articles in all Journals with MSC (2000): 54E45, 54F65, 57M30, 57N10

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: 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
Posted: 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 of article: Copyright 2006, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google