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

 

Inequivalent Cantor sets in $ R^{3}$ whose complements have the same fundamental group


Authors: Dennis J. Garity and Dušan Repovš
Journal: Proc. Amer. Math. Soc. 141 (2013), 2901-2911
MSC (2010): Primary 57M30, 57M05; Secondary 57N10, 54E45
Published electronically: April 10, 2013
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Abstract: For each Cantor set $ C$ in $ R^{3}$, all points of which have bounded local genus, we show that there are infinitely many inequivalent Cantor sets in $ R^{3}$ with the complement having the same fundamental group as the complement of $ C$. This answers a question from Open Problems in Topology and has as an application a simple construction of nonhomeomorphic open $ 3$-manifolds with the same fundamental group. The main techniques used are analysis of local genus of points of Cantor sets, a construction for producing rigid Cantor sets with simply connected complement, and manifold decomposition theory. The results presented give an argument that for certain groups $ G$, there are uncountably many nonhomeomorphic open $ 3$-manifolds with fundamental group $ G$.


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Additional Information

Dennis J. 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

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11911-8
PII: S 0002-9939(2013)11911-8
Keywords: Cantor set, rigidity, local genus, defining sequence, end, open $3$-manifold, fundamental group
Received by editor(s): November 2, 2011
Published electronically: April 10, 2013
Additional Notes: The authors were supported in part by the Slovenian Research Agency grants P1-0292-0101, J1-2057-0101 and BI-US/11-12-023. The first author was also supported in part by National Science Foundation grants DMS 0852030 and DMS 1005906
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2013 American Mathematical Society